Tensor Calculus and Differential Geometry

Home , Antisymmetric tensor, Differentiable manifold, Directional derivative, Holonomic basis, Metric tensor, Mixed tensor

Course Notes Tensor Calculus and Differential Geometry

2WAH0

Luc Florack

March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents

Preface iii

Notation 1

1 Prerequisites from Linear Algebra 3

2 Tensor Calculus 7

2.1 Vector Spaces and Bases ...... 7

2.2 Dual Vector Spaces and Dual Bases ...... 8

2.3 The Kronecker Tensor ...... 10

2.4 Inner Products ...... 11

2.5 Reciprocal Bases ...... 14

2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations ...... 16

2.7 Examples of Vectors and Covectors ...... 17

2.8 Tensors ...... 18

2.8.1 Tensors in all Generality ...... 18

2.8.2 Tensors Subject to Symmetries ...... 22

2.8.3 Symmetry and Antisymmetry Preserving Product Operators ...... 24

2.8.4 Vector Spaces with an Oriented Volume ...... 31

2.8.5 Tensors on an Inner Product Space ...... 34

2.8.6 Tensor Transformations ...... 36

2.8.6.1 “Absolute Tensors” ...... 37 CONTENTS i

2.8.6.2 “Relative Tensors” ...... 38

2.8.6.3 “Pseudo Tensors” ...... 41

2.8.7 Contractions ...... 43

2.9 The Hodge Star Operator ...... 43

3 Differential Geometry 47

3.1 Euclidean Space: Cartesian and Curvilinear Coordinates ...... 47

3.2 Differentiable Manifolds ...... 48

3.3 Tangent Vectors ...... 49

3.4 Tangent and Cotangent Bundle ...... 50

3.5 Exterior Derivative ...... 51

3.6 Afﬁne Connection ...... 52

3.7 Lie Derivative ...... 55

3.8 Torsion ...... 55

3.9 Levi-Civita Connection ...... 56

3.10 Geodesics ...... 57

3.11 Curvature ...... 58

3.12 Push-Forward and Pull-Back ...... 59

3.13 Examples ...... 60

3.13.1 Polar Coordinates in the Euclidean Plane ...... 61

3.13.2 A Helicoidal Extension of the Euclidean Plane ...... 63 ii CONTENTS Preface

These course notes are intended for students of all TU/e departments that wish to learn the basics of tensor calculus and differential geometry. Prerequisites are linear algebra and vector calculus at an introductory level. The treatment is condensed, and serves as a complementary source next to more comprehensive accounts that can be found in the (abundant) literature. As a companion for classroom adoption it does provide a reasonably self-contained introduction to the subject that should prepare the student for further self-study.

These course notes are based on course notes written in Dutch by Jan de Graaf. Large parts are straightforward translations. I am therefore indebted to Jan de Graaf for many of the good things. I have, however, taken the liberty to skip, rephrase, and add material, and will continue to update these course notes (the date on the cover reﬂects the version). I am of course solely to blame for the errors that might have been introduced in this undertaking.

Rigor is difﬁcult to reconcile with simplicity of terminology and notation. I have adopted Jan de Graaf’s style of presentation in this respect, by letting the merit of simplicity prevail. The necessary sacriﬁce of rigor is compensated by a great number of interspersed “caveats”, notational and terminological remarks, all meant to train the reader in coming to grips with the parlance of tensor calculus and differential geometry.

Luc Florack

Eindhoven, March 10, 2021. iv Notation

Instead of rigorous notational declarations, a non-exhaustive list of examples is provided illustrating the notation for the most important object types used in these course notes:

• Linear mappings: A , B, etc. • Matrices: A, B, etc.

• Tensors: A, B, etc. • Vectors: a, b, etc. • Covectors: aˆ, bˆ, etc.

i i • Basis vectors: ei, fi, etc. (for holonomic basis also ∂/∂x , ∂/∂y , etc.)

• Basis covectors: eˆi, ˆf i, etc. (for holonomic basis also dxi, dyi, etc.) • Vector components: ai, bi, etc.

• Covector components: ai, bi, etc.

• Tensor components: Ai1...ip , Bi1...ip , etc. j1...jq j1...jq 2 1. Prerequisites from Linear Algebra

Linear algebra forms the skeleton of tensor calculus and differential geometry. We recall a few basic deﬁnitions from linear algebra, which will play a pivotal role throughout this course.

Reminder A vector space V over the ﬁeld K (R or C) is a set of objects that can be added and multiplied by scalars, such that the sum of two elements of V as well as the product of an element in V with a scalar are again elements of V (closure property). Moreover, the following properties are satisﬁed for any u, v, w ∈ V and λ, µ ∈ K: • (u + v) + w = u + (v + w), • there exists an element o ∈ V such that o + u = u, • there exists an element −u ∈ V such that u + (−u) = o,

• u + v = v + u, • λ · (u + v) = λ · u + λ · v, • (λ + µ) · u = λ · u + µ · u,

• (λµ) · u = λ · (µ · u), • 1 · u = u. The inﬁx operator + denotes vector addition, the inﬁx operator · denotes scalar multiplication.

Terminology Another term for vector space is linear space.

Remark

• We will almost exclusively consider real vector spaces, i.e. with scalar ﬁeld K = R. • We will invariably consider ﬁnite-dimensional vector spaces.

Reminder n A basis of an n-dimensional vector space V is a set {ei}i=1, such that for each x ∈ V there exists a unique 1 n n Pn i n-tuple (x , . . . , x ) ∈ R such that x = i=1 x ei. 4 Prerequisites from Linear Algebra

Reminder

• A linear operator A : V → W , in which V and W are vector spaces over K, is a mapping that satisﬁes the following property: A (λ v + µ w) = λ A (v) + µ A (w) for all λ, µ ∈ K and v, w ∈ V . • The set L (V,W ) of all linear operators of this type is itself a vector space, with the following deﬁnitions of vector addition and scalar multiplication: (λA + µB)(v) = λA (v) + µB(v) for all λ, µ ∈ K, A , B ∈ L (V,W ) and v ∈ V .

Notation

The set of square n × n matrices will be denoted by Mn. Reminder

The determinant of a square matrix A ∈ Mn with entries Aij (i, j = 1, . . . , n) is given by

n n X 1 X det A = [j , . . . , j ] A ...A = [i , . . . , i ][j , . . . , j ] A ...A . 1 n 1j1 njn n! 1 n 1 n i1j1 injn j1,...,jn=1 i1, . . . , in = 1 j1, . . . , jn = 1

The completely antisymmetric symbol in n dimensions is deﬁned as follows:   +1 if (i1, . . . , in) is an even permutation of (1, . . . , n) , [i1, . . . , in] = −1 if (i1, . . . , in) is an odd permutation of (1, . . . , n) ,  0 otherwise .

Observation

n • Exactly n! of the n index realizations in [i1, . . . , in] yield nontrivial results. • To determine det A, compose all n! products of n matrix entries such that the row labels as well as the column labels of the factors in each product are all distinct. Rearrange factors such that the row labels are incrementally ordered. Assign a sign ±1 to each product corresponding to the permutation (even/odd) of the column labels observed in this ordered form, and add up everything. Cf. the following example.

Example Let A ∈ be a 3 × 3 matrix: M3   A11 A12 A13 A =  A21 A22 A23  . A31 A32 A33 Carrying out the recipe yields Table 1.1:

product with row labels ordered sign of column label permutation A11A22A33 [123] = +1 A12A23A31 [231] = +1 A13A21A32 [312] = +1 A13A22A31 [321] = −1 A11A23A32 [132] = −1 A12A21A33 [213] = −1

Table 1.1: det A = A11A22A33 +A12A23A31 +A13A21A32 −A13A22A31 −A11A23A32 −A12A21A33. Prerequisites from Linear Algebra 5

Observation

For a general square matrix A ∈ Mn the following identities hold:

n X [i1, . . . , in] Ai1j1 ...Ainjn = [j1, . . . , jn] det A,

i1,...,in=1 n X [j1, . . . , jn] Ai1j1 ...Ainjn = [i1, . . . , in] det A,

j1,...,jn=1 n n X X [i1, . . . , in][j1, . . . , jn] Ai1j1 ...Ainjn = n! det A.

i1,...,in=1 j1,...,jn=1

Products of completely antisymmetric symbols can be expressed in terms of determinants:   δi1j1 . . . δi1jn . . def i1...in [i1, . . . , in][j1, . . . , jn] = det  . .  = δ ,  . .  j1...jn

δinj1 . . . δinjn

and, more generally,   n δik+1jk+1 . . . δik+1jn X  . .  [i1, . . . , ik, ik+1, . . . , in][i1, . . . , ik, jk+1, . . . , jn] = k! det  . .  . i1,...,ik=1 δinjk+1 . . . δinjn

Example

• [i, j, k][`, m, n] = δi`δjmδkn + δimδjnδk` + δinδj`δkm − δimδj`δkn − δi`δjnδkm − δinδjmδk`. P3 • i=1 [i, j, k][i, m, n] = δjmδkn − δjnδkm. P3 • i,j=1 [i, j, k][i, j, n] = 2δkn. P3 • i,j,k=1 [i, j, k][i, j, k] = 6.

Caveat

One often writes det Aij instead of det A. The labels i and j are not free indices in this case!

Reminder ˜ ˜ij The cofactor matrix of A ∈ Mn is the matrix A ∈ Mn with entries A given by ∂ det A A˜ij = . ∂Aij

˜T The adjugate matrix of A ∈ Mn is a synonymous term for A ∈ Mn. Notation

Cofactor and adjugate matrix of A ∈ Mn are sometimes indicated by cofA and adjA, respectively. 6 Prerequisites from Linear Algebra

Notation i In any expression of the form Xi , with a matching pair of sub- and superscript i = 1, . . . , n, in which n = dim V is the dimension of an underlying vector space V , the Einstein summation convention applies, meaning that it is to be understood as an i-index-free sum: n i def X i Xi = Xi . i=1 This convention applies to each pair of matching sub- and superscript within the same index range. Indices to which the convention applies are referred to as spatial (or, depending on context, spatiotemporal) indices. Note that matching upper and lower indices are dummies that can be arbitrarily relabelled provided this does not lead to conﬂicting notation.

Observation

• Each term in the polynomial det A contains at most one factor Aij for each ﬁxed i, j = 1, . . . , n.

˜ij 2 ˜ij ∂A ∂ det A • As a result, A does not depend on Aij for ﬁxed i, j = 1, . . . , n: = 2 = 0. ∂Aij ∂Aij

d det A ∂ det A dAij ˜ij dAij • If A : R → Mn : t 7→ A(t) is a matrix-valued function, = = A (chain rule). dt ∂Aij dt dt

˜T ˜T ˜ik ˜ki i • AA = A A = det AI, or A adjA = adjAA = det AI, i.e. AjkA = A Akj = det A δj.

• If A is invertible, then A˜T = adjA = det AA−1.

ij • Each entry A˜ of A˜ = cofA is a homogeneous polynomial of degree n−1 in terms of the entries Aij of A.

• Unlike A−1, A˜ = cofA and A˜T = adjA are always well-deﬁned, even if A is singular. 2. Tensor Calculus

2.1. Vector Spaces and Bases

Ansatz

An n-dimensional vector space V over R furnished with a basis {ei}.

Notation Unless stated otherwise the Einstein summation convention applies to identical pairs of upper and lower indices. i i def Pn i Thus if Xj is any collection of numbers, then Xi = i=1 Xi .

Terminology The act of constraining one upper and one lower index to be equal—and thus tacitly assuming summation, v.s.—is called an index contraction, or a trace.

Remark Contracted indices are dummies that can be arbitrarily relabeled, as opposed to free indices. Thus for example i j Xi = Xj .

Remark i For each basis {ei} of V and each x ∈ V there exists a unique collection of real numbers x such that i x = x ei, recall chapter 1.

Terminology The real numbers xi are referred to as the contravariant components of the vector x relative to the vector basis {ei}.

Deﬁnition 2.1 The Kronecker symbol, or Kronecker-δ, is deﬁned as 1 if i = j δi = j 0 if i 6= j .

i i i i Theorem 2.1 Let x = x ei ∈ V and fj = Ajei deﬁne a change of basis. If x ei = y fi, then i i j i i j x = Ajy or, equivalently, y = Bjx , i k i in which, by deﬁnition, AkBj = δj.

Caveat i Pay attention to the occurrence of the components Aj in the transformation of basis vectors ei and contravariant components xi, respectively. 8 Tensor Calculus

Observation Given a smooth manifold—to be defined later—with a coordinate system xi , the collection of linear partial i derivative operators ei ≡ ∂i ≡ ∂/∂x at any (implicit) fiducial point defines a vector space basis. Upon a coordinate transformation xi = xi(xj) the chain rule relates the corresponding bases by

i ∂j = Aj∂i ,

j in which fj ≡ ∂j ≡ ∂/∂x , and ∂xi Ai = . j ∂xj This is formally identical to the basis transformation in Theorem 2.1 on page 7.

Notation i On a smooth manifold one often encounters vector decomposition relative to a coordinate basis, i.e. v = v ∂i. (The ﬁducial point is implicit here.)

Terminology

The set {∂i} is called a coordinate (vector) basis, or a holonomic (vector) basis. By deﬁnition it spans the tangent space of the manifold at the ﬁducial point.

Reminder

• Basis vectors are labeled with lower indices. • Contravariant vector components are labeled with upper indices.

Remark The distinction into upper and lower indices is relevant only if the indices are intrinsically constrained to take values in the range {1, . . . , n}, where n = dim V .

2.2. Dual Vector Spaces and Dual Bases

Ansatz An n-dimensional vector space V over R.

Deﬁnition 2.2 A linear functional on V is a linear mapping from V into R. In other words, ˆf : V → R is a linear functional if for all x, y ∈ V , λ, µ ∈ R, ˆf(λx + µy) = λˆf(x) + µˆf(y) .

Remark The set V ∗ consisting of all linear functionals on V can be turned into a vector space in the usual way, viz. let ˆf, gˆ ∈ V ∗ and λ, µ ∈ R, deﬁne the linear combination λˆf + µgˆ ∈ V ∗ by virtue of the identiﬁcation (λˆf + µgˆ)(x) = λˆf(x) + µgˆ(x)

for all x ∈ V , recall chapter 1.

Deﬁnition 2.3 The dual space is the vector space of linear functionals on V : V ∗ = L (V, R). 2.2 Dual Vector Spaces and Dual Bases 9

Ansatz ∗ An n-dimensional vector space V over R furnished with a basis {ei}, and its dual space V .

Remark i ∗ i i i Each basis {ei} of V induces a basis eˆ of V as follows. If x = x ei, then eˆ (x) = x .

Result 2.1 The dual basis eˆi of V ∗ is uniquely determined through its deﬁning property

i i eˆ (ej) = δj .

Proof of Result 2.1. Let ˆf ∈ V ∗.

i ˆ ˆ i ˆ i ˆ i ˆ i i • For any x = x ei ∈ V we have f(x) = f(x ei) = f(ei)x = f(ei)eˆ (x) = (f(ei)eˆ )(x) = (fieˆ )(x) ˆ i ∗ with fi = f(ei). This shows that the set {eˆ } spans V .

i ∗ j j j • Suppose αieˆ = 0 ∈ V , then αi = αjδi = αjeˆ (ei) = (αjeˆ )(ei) = 0 ∈ R for all i = 1, . . . , n. This shows that {eˆi} is a linearly independent set.

Assumption

In view of this result, it will henceforth be tacitly understood that a given vector space V (with basis {ei}) implies the existence of its dual space V ∗ (with corresponding dual basis eˆi ).

Observation The dual basis V ∗ has the same dimension as V . Terminology A linear functional ˆf ∈ V ∗ is alternatively referred to as a covector or 1-form. The dual space V ∗ of a vector space V is also known as the covector space or the space of 1-forms.

Remark i ∗ ∗ For each covector basis eˆ of V and each xˆ ∈ V there exists a unique collection of real numbers {xi} such i that xˆ = xieˆ .

Terminology i The real numbers xi are referred to as the covariant components of the covector xˆ relative to covector basis eˆ .

We end this section with the dual analogue of Theorem 2.1 on page 7:

i Theorem 2.2 The change of basis fj = Ajei induces a change of dual vector basis

ˆj j i f = Bi eˆ , k i i i ˆi ∗ in which Aj Bk = δj. Consequently, if xˆ = xieˆ = yif ∈ V , then

j j xi = Bi yj or, equivalently, yi = Ai xj .

Proof of Theorem 2.2.

` ˆi i k i ˆi i k ` i ` k i ` k • Given fj = Aje` and f = Bkeˆ we obtain δj = f (fj) = Bkeˆ (Aje`) = BkAjeˆ (e`) = BkAjδ` = i k −1 BkAj , from which it follows that B = A . 10 Tensor Calculus

i ˆi ˆj j i j • Thus if we deﬁne xieˆ = yif , then yjf = yjBi eˆ , whence xi = Bi yj. • Inversion yields the last identity.

Terminology

• Theorem 2.1, page 7, deﬁnes the so-called vector transformation law. • Theorem 2.2, page 9, deﬁnes the so-called covector transformation law.

Observation Recall the observation on page 8. Given a smooth manifold with a coordinate system xi , and a coordinate transformation xi = xi(xj), the chain rule (at any ﬁducial point) states that

∂xj dxj = Bjdxi in which Bj = . i i ∂xi

Identifying eˆi ≡ dxi and ˆf j ≡ dxj, this is formally identical to the dual vector basis transformation in Theorem 2.2. In particular we have, recall Result 2.1,

i i dx (∂j) = δj .

Notation i On a smooth manifold one often encounters covector decomposition relative to a coordinate basis, i.e. vˆ = vidx . (The ﬁducial point is implicit here.)

Terminology The set {dxi} is called a coordinate (dual vector) basis, or holonomic (dual vector) basis. By deﬁnition it spans the cotangent space of the manifold at the ﬁducial point.

Reminder

• Basis covectors are labeled with upper indices. • Covariant components are labeled with lower indices.

2.3. The Kronecker Tensor

Ansatz An n-dimensional vector space V over R. Notation Instead of a standard function notation for ˆf ∈ V ∗ one often employs the bracket formalism hˆf, · i ∈ V ∗. Function evaluation is achieved by dropping a vector argument x ∈ V into the empty slot: hˆf, xi = ˆf(x) ∈ R. In particular we have i i i heˆ , eji = eˆ (ej) = δj for dual bases, recall Result 2.1 on page 9.

∗ Deﬁnition 2.4 The bilinear map h · , · i : V × V → R :(ˆf, x) 7→ hˆf, xi is called the Kronecker tensor. 2.4 Inner Products 11

Remark The general concept of a tensor will be explained in section 2.8.

Caveat Do not confuse the Kronecker tensor with an inner product. Unlike the Kronecker tensor an inner product requires two vector arguments from the same vector space.

Observation The Kronecker tensor is bilinear:

∗ •∀ ˆf, gˆ ∈ V , x ∈ V, λ, µ ∈ R : hλˆf + µgˆ, xi = λhˆf, xi + µhgˆ, xi,

∗ •∀ ˆf ∈ V , x, y ∈ V, λ, µ ∈ R : hˆf, λx + µyi = λhˆf, xi + µhˆf, yi. • Result 2.1 on page 9 shows that the components of the Kronecker tensor are given by the Kronecker symbol. ˆ i ∗ i ˆ • If f = fieˆ ∈ V and x = x ei ∈ V , then bilinearity of the Kronecker tensor implies hf, xi = j i j i i fix heˆ , eji = fix δj = fix . Thus Kronecker tensor and index contraction are close-knit.

Remark

• The components of the Kronecker tensor are independent of the choice of basis. (Verify!) • Given V , the assumption on page 9 implies that the Kronecker tensor exists.

• Since V ∗ is itself a vector space one may consider its dual (V ∗)∗ = V ∗∗. Without proof we state that if V (and thus V ∗) is finite-dimensional, then V ∗∗ is isomorphic to, i.e. may be identified with V . This does not generalize to infinite-dimensional vector spaces.

Notation The use of the formally equivalent notations ˆf and hˆf, · i for an element of V ∗ reflects our inclination to regard it either as an atomic entity, or as a linear mapping with unspecified argument. The previous remark justifies the use of similar alternative notations for an element of V , viz. either as x, say, or as h · , xi ∈ V ∗∗ ∼ V .

2.4. Inner Products

Ansatz An n-dimensional vector space V over R.

Deﬁnition 2.5 An inner product on a real vector space V is a nondegenerate positive symmetric bilinear mapping of the form ( · | · ): V × V → R which satisﬁes

•∀ x, y ∈ V :(x|y) = (y|x),

Deﬁnition 2.5 will be our default. In the context of smooth manifolds it is referred to as a Riemannian inner product or, especially in physics, as a ‘Riemannian metric’. Occasionally we may encounter an inner product on a complex vector space. Below, z = a − bi ∈ C denotes the complex conjugate of z = a + bi ∈ C. 12 Tensor Calculus

Deﬁnition 2.6 An inner product on a complex vector space V is a nondegenerate positive hermitean sesquilinear mapping of the form ( · | · ): V × V → C which satisﬁes

•∀ x, y ∈ V :(x|y) = (y|x),

Caveat

• Note that for the complex case (λx + µy|z) = λ (x|z) + µ (y|z), whence the attribute ‘sesquilinear’. • The term “inner product” is used in different contexts to denote bilinear or sesquilinear forms satisfying different sets of axioms. Examples include the Lorentzian inner product in the context of special relativity (v.i.) and the symplectic inner product in the Hamiltonian formalism of classical mechanics. • Unlike with the Kronecker tensor, the existence of an inner product is never implied, but must always be stipulated explicitly.

Deﬁnition 2.7 A pseudo-Riemannian inner product on a real vector space V is a nondegenerate symmetric bilinear mapping of the form ( · | · ): V × V → R which satisﬁes

•∀ x, y ∈ V :(x|y) = (y|x),

•∀ x, y, z ∈ V, λ, µ ∈ R :(x|λy + µz) = λ (x|y) + µ (x|z), •∀ x ∈ V with x 6= 0 ∃ y ∈ V such that (x|y) 6= 0.

A Lorentzian inner product is a pseudo-Riemannian inner product with signature (+, −,..., −) or (−, +,..., +).

Note that Definition 2.5 is a special case of Definition 2.7, viz. with signature (+,..., +). The signature of a Lorentzian inner product contains exactly one + or one − sign. The notion of signature is explained below in Definition 2.8. Terminology A vector space V endowed with an inner product ( · | · ) is also called an inner product space.

Ansatz

An n-dimensional inner product space V over R furnished with a basis {ei}.

Deﬁnition 2.8 The Gram matrix G is the matrix with components gij = (ei|ej). Its signature (s1, . . . , sn) is deﬁned as the (basis independent) n-tuple of signs si = ±, i = 1, . . . , n, of the (real) eigenvalues of the Gram matrix for a given ordering of its eigenvectors.

Remark Clearly, G is symmetric. Another consequence of the definition is that G is invertible in the pseudo-Riemannian i i case. For suppose G would be singular, then there exists x = x ei ∈ V , x 6= 0, such that x gij = 0. As a result i i j i j i j we find that for all y = y ei ∈ V , 0 = x gijy = x (ei|ej) y = x ei|y ej = (x|y). This contradicts the non-degeneracy axiom of Definition 2.7. 2.4 Inner Products 13

Terminology The components of the inverse Gram matrix G−1 are denoted by gij, i.e.

ik i g gkj = δj .

An inner product, if it exists, can be related to the Kronecker tensor, as we will see below.

Ansatz An n-dimensional inner product space V over R.

Theorem 2.3 There exists a bijective linear map G : V → V ∗, with inverse G−1 : V ∗ → V , such that

•∀ x, y ∈ V :(x|y) = hG(x), yi,

•∀ xˆ ∈ V ∗, y ∈ V : G−1(xˆ)|y = hxˆ, yi.

The matrix representations of G and G−1 are given by the Gram matrix G and its inverse G−1, respectively, recall Deﬁnition 2.8 on page 12.

Proof of Theorem 2.3.

• For fixed x ∈ V define xˆ ∈ V ∗ as the linear map xˆ : V → R : y 7→ (x|y). Subsequently define the linear map G : V → V ∗ : x 7→ G(x) ≡ xˆ.

• For the second part we need to prove invertibility of G. Suppose x ∈ V, x 6= 0, is such that G(x) = 0. Then 0 = hG(x), yi = (x|y) for all y ∈ V , contradicting the nondegeneracy property of the inner product. Hence G is one-to-one1. Since dim V = n < ∞ it is also bijective, hence invertible.

i i i • To compute the matrix representation G of G, decompose x = x ei, y = y ei, and G(x) = G(x ei) = i j j x Gijeˆ , with matrix components deﬁned via G(ei) = Gijeˆ .

– By construction (first item above) and by virtue of bilinearity of the inner product and the definition i j i j of the Gram matrix, we have hG(x), yi = (x|y) = x y (ei|ej) = gijx y . i k j i k j i j – By definition of duality, we have hG(x), yi = Gijx y heˆ , eki = Gijx y δk = Gijx y .

i j Since x , y ∈ R are arbitrary it follows that Gij = gij, recall Deﬁnition 2.8.

• Consequently, the inverse G−1 : V ∗ → V has a matrix representation G−1 corresponding to the inverse Gram matrix: (G−1)ij = gij.

Remark

• Note the synonymous formulations: G(x) ≡ xˆ ≡ hG(x), · i ≡ (x| · ). Recall the comment on notation, page 11. Deﬁnition 2.9 below introduces yet another synonymous form. • The Riesz representation theorem, which states that for each xˆ ∈ V ∗ there exists one and only one x ∈ V such that hxˆ, yi = (x|y) for all y ∈ V , is an immediate consequence of Theorem 2.3, viz. x = G−1(xˆ).

1The terms “one-to-one” and “injective” are synonymous. A function f : D → C is injective if it associates distinct arguments with distinct values. If the codomain C of the function is replaced by its actual range f(D), then it is also surjective, and hence bijective. 14 Tensor Calculus

Notation

• We reserve boldface symbols for intrinsic quantities such as vectors, covectors, and (other) linear mappings, in order to distinguish them from their respective components relative to a ﬁducial basis.

Deﬁnition 2.9 The conversion operators ] : V → V ∗ and [ : V ∗ → V are introduced as shorthand for G, respectively G−1, viz. if v ∈ V , vˆ ∈ V ∗, then

]v def= G(v) ∈ V ∗ respectively [vˆ def= G−1(vˆ) ∈ V.

Terminology G is commonly known as the metric, and G−1 as the dual metric. The operators ] : V → V ∗ and [ : V ∗ → V are pronounced as sharp and ﬂat, respectively.

Caveat In the literature, the roles of sharp and ﬂat are often reversed! Make sure to adhere to an unambiguous convention.

i i ∗ Result 2.2 Let v = v ei ∈ V and vˆ = vieˆ ∈ V , then

i j j ∗ i • ]v = gijv eˆ = vjeˆ = vˆ ∈ V , with vj = gijv ;

ij j j ij • [vˆ = g viej = v ej = v ∈ V , with v = g vi.

Proof of Result 2.2. This follows directly from the deﬁnition of ] and [ using

j ∗ • ]ei = G(ei) = gijeˆ ∈ V , respectively

i −1 i ij • [eˆ = G (eˆ ) = g ej ∈ V .

Notation i def i Unless stated otherwise we will adhere to the following convention. If v ∈ R, then vj = gijv . reversely, if j def ij vi ∈ R, then v = g vi. Caveat In the foregoing G ∈ L (V,V ∗) and G−1 ∈ L (V ∗,V ), with prototypes G : V → V ∗, resp. G−1 : V ∗ → V . However, one often interprets the metric and its dual as bilinear mappings of type G : V × V → R, resp. G−1 : V ∗ × V ∗ → R. It should be clear from the context which prototype is implied. We return to this ambiguity in section 2.8.

Reminder In an inner product space vectors can be seen as “covectors in disguise”, vice versa.

2.5. Reciprocal Bases

Ansatz

An n-dimensional inner product space V over R furnished with a basis {ei}. 2.5 Reciprocal Bases 15

i i ij Deﬁnition 2.10 The basis {ei} of V induces a new basis [eˆ of V given by [eˆ = g ej, the so-called reciprocal basis.

i ∗ ∗ j Deﬁnition 2.11 The basis eˆ of V induces a new basis {]ei} of V given by ]ei = gijeˆ , the so-called reciprocal dual basis.

Caveat

i i • Instead of [eˆ one sometimes writes e for the reciprocal of ei.

i • Likewise, instead of ]ei one sometimes writes eˆi for the reciprocal of eˆ . • Since we conventionally label elements of V by lower indices, and elements of V ∗ by upper indices, we will adhere to the notation employing the [ and ] operator.

Remark

• The reciprocal basis is uniquely determined by the inner product.

−1 i j ik j` • The Gram matrix of the reciprocal basis is G , with components [eˆ |[eˆ = g g (ek|e`) = ik j` ij g g gk` = g .

i ik ik i • We have [eˆ |ej = g (ek|ej) = g gkj = δj.

Terminology i The vectors [eˆ and ej are said to be perpendicular for i 6= j.

Remark i For each reciprocal basis [eˆ of V and each x ∈ V there exists a unique collection of real numbers {xi} such i that x = xi[eˆ .

Terminology

The real numbers xi are referred to as the covariant components of the vector x relative to the basis {ei}.

Reminder

• Reciprocal basis vectors are labeled with upper indices. • Covariant components are labeled with lower indices. • No confusion is likely to arise if we adhere to the notation using the [ and ] converters. 16 Tensor Calculus

2.6. Bases, Dual Bases, Reciprocal Bases: Mutual Relations

Heuristics Dual and reciprocal bases are close-knit. The exact relationship is clariﬁed in this section.

Sofar, a vector space V and its associated dual space V ∗ have been introduced as a priori independent entities. V and V ∗ are mutually connected by virtue of the “interaction” between their respective elements, recall Result 2.1 on page 9. One nevertheless retains a clear distinction between a vector (element of V ) and a dual vector (element of V ∗) throughout. That is, there is no mechanism to convert a vector into a dual vector, vice versa.

An inner product provides us with an explicit mechanism to construct a one-to-one linear mapping (i.e. dual vector) associated with each vector by virtue of the metric G, cf. Theorem 2.3 on page 13. An inner product also enables us to construct a reciprocal basis that allows us to mimic duality within V proper. “Interchangeability” of vectors and reciprocal vectors as well as that of vectors and dual vectors is governed by the Gram matrix G and its inverse. Ansatz

An n-dimensional inner product space V over R furnished with a basis {ei}.

Observation ∗ i Result 2.3 on page 13 states that each vector x ∈ V induces a dual vector xˆ = G(x) ∈ V . Let x = x ei 2 i i ∗ relative to basis {ei} of V . Then xˆ =x îeˆ relative to the corresponding dual basis {eˆ } of V . According to i i i Definition 2.10 on page 15 we also have x = xi[eˆ relative to the reciprocal basis {[eˆ } of V . Given x ∈ R we j have xi =x î = gijx .

Caveat Notice the distinct prototypes of metric G : V → V ∗ and linear map G : V → V represented by the Gram i matrix G, i.e. ei = G ([eˆ ).

Notation

No confusion is likely to arise if we use identical symbols, xî ≡ xi, to denote both dual as well as reciprocal vector components. The underlying basis ({eî}, respectively {[eî}), and thus the correct geometric interpretation, should be inferred from the context. By default we shall interpret xi as dual vector components, henceforth i ∗ omitting theˆsymbol, i.e. xˆ = xieˆ ∈ V .

2 For the sake of argument we temporarily write xˆi for dual, and xi for reciprocal vector components. 2.7 Examples of Vectors and Covectors 17

Reminder

• A dual basis does not require an inner product, a reciprocal basis does.

• A dual basis spans V ∗, a reciprocal basis spans V . • In an inner product space—with metric G—the components of a vector x ∈ V relative to the reciprocal basis are identical to the components of the dual vector xˆ = G(x) relative to the dual basis. • Thus, in an inner product space with ﬁxed metric G, one may loosely identify a vector x ∈ V and its dual vector xˆ ∈ V ∗ to the extent that xˆ = G(x) and ( · |x) = hxˆ, · i. • The metric and its dual, G, respectively G−1, provide the natural mechanism to “toggle” between elements of V and V ∗. Abbreviations: G(x) = ]x, G−1(xˆ) = [xˆ.

• The Gram matrix and its inverse, G, respectively G−1, or rather the associated linear mappings G : V → V and G −1 : V → V , provide natural mechanisms to “toggle” between elements of V and their reciprocals in V .

2.7. Examples of Vectors and Covectors

It is important to learn to recognize covectors and vectors in practice, and to maintain a clear distinction throughout. Some important examples from mechanics may illustrate this.

Example

• Particle velocity v provides an example of a vector. • Particle momentum pˆ provides an example of a covector.

• The metric tensor allows us to relate these two: pˆ = G(v) = ]v, or equivalently, v = G−1(pˆ) = [pˆ.

Example

• Displacement x relative to a ﬁducial point in a continuous medium (or in vacuo) provides an example of a vector. • Spatial frequency kˆ provides an example of a covector.

• The Kronecker tensor allows us to construct a scalar from these two: hkˆ, xi ∈ R (“phase”).

Caveat

• In practice one often refrains from a notational distinction between covectors and vectors, writing e.g. p and k instead of pˆ and kˆ, as we do here.

• By the same token it is a common malpractice to speak of “the momentum vector p” and the “wave vector k” even if covectors are intended. (Of course, technically speaking covectors are vectors.) 18 Tensor Calculus

2.8. Tensors

2.8.1. Tensors in all Generality

Ansatz

• An n-dimensional vector space V over R.

• A basis {ei} of V .

∗ ∗ Deﬁnition 2.12 A tensor is a multilinear mapping of type T : V ×... × V ×V ×... × V → R, with p, q ∈N0. | {z } | {z } p q

∗ ∗ def p We collectively denote tensors of this type as follows: T ∈ V ⊗ ... ⊗ V ⊗ V ⊗ ... ⊗ V = Tq (V ). | {z } | {z } p q

Caveat

• The ordering of arguments matters! Our default ordering will be that of Deﬁnition 2.12. • Thus the prototype may be speciﬁed with a different arrangement of linear spaces V and V ∗. • Without default ordering, the explicit ⊗-notation is the more informative one.

Observation

p • Tq (V ) constitutes a linear space subject to the usual rules for linear superposition of mappings, i.e. for all p S, T ∈ Tq (V ) and all λ, µ ∈ R we declare (λS + µT)((co)vectors) = λS((co)vectors) + µT((co)vectors) ,

in which “(co)vectors” generically denotes an appropriate sequence of vector and/or covector arguments.

p • Recall that the linear structure of Tq (V ) has nothing to do with that of its domain of deﬁnition, but is induced by the linear structure on its codomain, in this case the trivial vector space R (cf. the r.h.s. above). Terminology

p p • We refer to T ∈ Tq (V ) sometimes as a q -tensor.

p • T ∈ Tq (V ) is referred to as a mixed tensor of contravariant rank p and covariant rank q. 0 • If q 6= 0, T ∈ Tq(V ) is called a covariant tensor of rank q. Covariant tensors take contravariant arguments, i.c. vectors.

p • If p 6= 0, T ∈ T0(V ) is called a contravariant tensor of rank p. Contravariant tensors take covariant arguments, i.c. covectors.

0 • T ∈ T0(V ) is called a scalar. We identify T with its value in this case, i.e. T ∈ R. • The words cotensor/contratensor are synonymous to covariant/contravariant tensor. 2.8 Tensors 19

Example 1 ∗ Let T be a 2 -tensor, and x, x1, x2, y, y1, y2 ∈ V , zˆ, zˆ1, zˆ2 ∈ V , and λ1, λ2 ∈ R arbitrary. Then we have

• T(λ1zˆ1 + λ2zˆ2, x, y) = λ1T(zˆ1, x, y) + λ2T(zˆ2, x, y);

• T(zˆ, λ1x1 + λ2x2, y) = λ1T(zˆ, x1, y) + λ2T(zˆ, x2, y);

• T(zˆ, x, λ1y1 + λ2y2) = λ1T(zˆ, x, y1) + λ2T(zˆ, x, y2).

Result 2.3 Special cases of low rank tensors:

0 0 • A 0 -tensor is a scalar, i.e. T0(V ) = R.

0 0 ∗ • A 1 -tensor is a covector, i.e. T1(V ) = V .

1 1 • A 0 -tensor is a vector, i.e. T0(V ) = V .

Proof of Result 2.3. The ﬁrst case is a convention previously explained. The second case is true by Deﬁnition 2.2, page 8, recall the terminology introduced on page 9. The third case is true by virtue of the isomorphism V ∗∗ = V , recall the remark on page 11.

Deﬁnition 2.13 The outer product f ⊗ g : X × Y → R of two R-valued functions f : X → R and g : Y → R is deﬁned as follows: (f ⊗ g)(x, y) = f(x)g(y) for all x ∈ X, y ∈ Y .

Since vectors and covectors are R-valued mappings, the following deﬁnition is a natural one.

p ˆj1 ˆjq p Deﬁnition 2.14 The mixed q -tensor ei1 ⊗...⊗eip ⊗e ⊗...⊗e ∈ Tq (V ) is deﬁned, for each (p+q)-tuple of indices (i1, . . . , ip, j1, . . . , jq), by

j1 jq j1 jq ei1 ⊗...⊗ eip ⊗ eˆ ⊗...⊗ eˆ (xˆ1,. . ., xˆp, y1,. . ., yq) = hxˆ1, ei1 i...hxˆp, eip iheˆ , y1i...heˆ , yqi ,

∗ for all xˆ1,. . ., xˆp ∈ V and y1,. . ., yq ∈ V .

Caveat

• The symbol ⊗ used in Deﬁnitions 2.13 and 2.14 should not be confused with the one used in Deﬁnition 2.12.

j1 jq ∗ ∗ • Mnemonic: ei1 ⊗...⊗ eip ⊗ eˆ ⊗...⊗ eˆ ∈ V ⊗ ... ⊗ V ⊗ V ⊗ ... ⊗ V . | {z } | {z } | {z } | {z ∗ } p vectors q covectors p copies of V q copies of V

i ...i i ...i Result 2.4 If T∈Tp(V ), then there exist numbers t 1 p ∈ such that T = t 1 p e ⊗...⊗e ⊗eˆj1 ⊗...⊗eˆjq , q j1...jq R j1...jq i1 ip i ...i viz. t 1 p = T(eˆi1 ,..., eˆip , e ,..., e ). j1...jq j1 jq 20 Tensor Calculus

Proof of Result 2.4. We have

i1 ip T eˆ ,..., eˆ , ej1 ,..., ejq k ...k = t 1 p e ⊗ ... ⊗ e ⊗ eˆ`1 ⊗ ... ⊗ eˆ`q eî1 ,..., eîp , e ,..., e `1...`q k1 kp j1 jq k ...k = t 1 p heî1 , e i...heîp , e iheˆ`1 , e i...heˆ`q , e i `1...`q k1 kp j1 jq k ...k i ` i ...i = t 1 p δi1 . . . δ p δ`1 . . . δ q = t 1 p . `1...`q k1 kp j1 jq j1...jq

∗ By multilinearity we have, for general covector and vector arguments ωˆ 1,..., ωˆ p ∈V , v1,..., vq ∈V ,

i ...i T (ωˆ ,..., ωˆ , v ,..., v ) = t 1 p ω . . . ω vj1 . . . vjq . 1 p 1 q j1...jq 1,i1 p,ip 1 q

Terminology

0 ∗ i • If T ∈ T1(V ) = V there exist numbers ti ∈ R such that T = tieˆ , viz. ti = T(ei). These are the covariant components of T relative to basis {ei}.

1 i i i i • If T ∈ T0(V ) = V there exist numbers t ∈ R such that T = t ei, viz. t = T(eˆ ). These are the contravariant components of T relative to basis {ei}.

• If T ∈ Tp(V ), the numbers ti1...ip ∈ are coined the mixed components of T relative to basis {e }. q j1...jq R i

Caveat i ...i Laziness has led to the habit of saying that T = t 1 p e ⊗ ... ⊗ e ⊗ eˆj1 ⊗ ... ⊗ eˆjq is expressed “relative to j1...jq i1 ip j1 jq basis {ei}”, where it would have been more accurate to refer to the actual basis {ei1 ⊗...⊗eip ⊗eˆ ⊗...⊗eˆ }.

Example Recall Deﬁnition 2.4 on page 10. We have yet another notation for the Kronecker tensor h · , · i ∈ V ⊗ V ∗, viz.

i i j h · , · i = ei ⊗ eˆ = δjei ⊗ eˆ , in which we recognize the Kronecker symbol as the mixed components of the Kronecker tensor relative to basis i ∗ i {ei}. To see this, insert an arbitrary covector yˆ = yieˆ ∈ V and vector x = x ei ∈ V on the r.h.s. The result is

i i k j i k j i j i j i ei ⊗ eˆ (yˆ, x) = hyˆ, eiiheˆ , xi = yjx heˆ , eiiheˆ , eki = yjx δi δk = yix δj = yix heˆ , eji = hyˆ, xi .

Note that the Kronecker tensor is parameter free. For this reason it is sometimes called a constant tensor. Since i its coefﬁcients δj do not depend on the chosen basis (an untypical property for tensor coefﬁcients, as we will see in Section 2.8.6), they are said to constitute an invariant holor.

Observation

p p+q • We have dim Tq (V ) = n .

• Lexicographical ordering of the components ti1...ip of tensors T ∈ Tp(V ) establishes an isomorphism j1...jq q p+q with the “column vector space” Rn . 2.8 Tensors 21

Terminology

• The indexed collection of tensor coefﬁcients ti1...ip is also referred to as a holor. j1...jq • Physicists and engineers are wont to refer to holors as “tensors”. In a mathematician’s vocabulary the (basis dependent) holor is strictly to be distinguished from the associated (basis independent) tensor, i ...i T = t 1 p e ⊗ ... ⊗ e ⊗ eˆj1 ⊗ ... ⊗ eˆjq . j1...jq i1 ip

p j1 jq • If {ei} is a basis of V , a basis of Tq (V ) is given by {ei1 ⊗ ... ⊗ eip ⊗ eˆ ⊗ ... ⊗ eˆ }. But do recall the caveat on page 20 concerning terminology.

Caveat Parlance and notation in tensor calculus tend to be somewhat sloppy. This is the price one must pay for simplicity. It is up to the reader to resolve ambiguities, if any, and provide correct mathematical interpretations.

Example It is customary to think of the metric G ∈ L (V,V ∗) as a covariant tensor of rank 2. In order to understand this, ∗ ∗ 0 let us clarify a previous remark on page 14. For given metric G, deﬁne g ∈ V ⊗ V = T2(V ) as follows:

g : V × V → R :(v, w) 7→ g(v, w) ≡ (G(v)) (w) ≡ (v|w) ≡ h]v, wi , recall the sharp operator, introduced on page 14.

Notation

• Except when this might lead to confusion, no notational distinction will henceforth be made between such distinct prototypes as g and G. We will invariably write G, whether it is intended as a covariant 2-tensor (g in the example, implied when we write G(v, w)), or as a vector-to-covector converter (]-operator, implied when we write G(v)). • The ]- and [-operators provide convenient shorthands for the (dual) metric if one wishes to stress its action as a vector-to-covector (respectively covector-to-vector) converter.

Observation

• On an inner product space the ]- and [-operators may be invoked implicitly to convert arguments to the prototype that is compatible with each tensor slot. We return to this in Section 2.8.5.

Terminology Irrespective of the intended interpretation, G (or g) will be referred to as the metric tensor.

p r Theorem 2.4 If S ∈ Tq (V ), T ∈ Ts(V ), with

i ...i S = s 1 p e ⊗ ... ⊗ e ⊗ eˆj1 ⊗ ... ⊗ eˆjq and T = ti1...ir e ⊗ ... ⊗ e ⊗ eˆj1 ⊗ ... ⊗ eˆjs , j1...jq i1 ip j1...js i1 ir

i ...i i ...i then S ⊗ T ∈ Tp+r(V ), with S ⊗ T = s 1 p t p+1 p+r e ⊗ ... ⊗ e ⊗ eˆj1 ⊗ ... ⊗ eˆjq+s . q+s j1...jq jq+1...jq+s i1 ip+r

Terminology p r p+r For S ∈ Tq (V ), T ∈ Ts(V ), the tensor S ⊗ T ∈ Tq+s(V ) is also referred to as the outer product or tensor product of S and T. 22 Tensor Calculus

Caveat Notice that we have chosen to adhere to a default ordering of basis and dual basis vectors that span the tensor product space, recall Deﬁnition 2.12, page 18. This boils down to a rearrangement such that the ﬁrst p + r arguments of the product tensor S ⊗ T are covectors, and the remaining q + s arguments are vectors.

Proof of Theorem 2.4. Apply Deﬁnition 2.13, page 19, accounting for the occurrence of multiplicative coefﬁ- cients: ((λf) ⊗ (µg))(x, y) = ((λf)(x))((µg)(y)) = λµf(x)g(y). Further details are left to the reader (taking into account the previous caveat).

Caveat The tensor product is associative, distributive with respect to addition and scalar multiplication, but not commu- tative. (Work out the exact meaning of these statements in formulae.)

Reminder

p • A mixed tensor T ∈ Tq (V ) has contravariant rank p and covariant rank q.

• The holor ti1...ip ∈ of T ∈ Tp(V ) is labeled with p upper and q lower indices. j1...jq R q

j1 jq p • Elements of the underlying basis {ei1 ⊗ ... ⊗ eip ⊗ eˆ ⊗ ... ⊗ eˆ } of Tq (V ), on the other hand, are labeled with p lower and q upper indices. • A holor depends on the chosen basis, the tensor itself (by deﬁnition!) does not. (We return to this aspect in Section 2.8.6.) • The outer product of two tensors yields another tensor. Ranks add up.

2.8.2. Tensors Subject to Symmetries

Ansatz

• An n-dimensional vector space V over R.

• A basis {ei} of V .

p • Tensor spaces of type Tq (V ).

p Certain subspaces of Tq (V ) are of special interest. For instance, it may be the case that the ordering of (co)vector arguments in a tensor evaluation does not affect the outcome.

Remark Recall a previous remark on page 8 concerning the convention for index positioning. In particular, in Deﬁni- tions 2.15 and 2.16 below, indices are used to label an a priori unconstrained number of arguments, whence the upper/lower index convention is not applicable.

Notation

• π = {π(α1), . . . , π(αp)} denotes a permutation of the p ordered symbols {α1, . . . , αp}.

• π(αi), i = 1, . . . , p, is shorthand notation for the i-th entry of π. 2.8 Tensors 23

0 Deﬁnition 2.15 Let π = {π(1), . . . , π(p)} be any permutation of labels {1, . . . , p}, then T ∈ Tp(V ) is called a symmetric covariant tensor if for all v1,..., vp ∈ V

T(vπ(1),..., vπ(p)) = T(v1,..., vp) .

p ∗ Likewise, T ∈ T0(V ) is called a symmetric contravariant tensor if for all vˆ1,..., vˆp ∈ V

T(vˆπ(1),..., vˆπ(p)) = T(vˆ1,..., vˆp) .

Notation

W 0 • The vector space of symmetric covariant p-tensors is denoted p(V ) ⊂ Tp(V ). Wp p • The vector space of symmetric contravariant p-tensors is denoted (V ) ⊂ T0(V ). W ∗ ∗ Wp • Alternatively one sometimes writes p(V ) = V ⊗S ... ⊗S V and (V ) = V ⊗S ... ⊗S V . | {z } | {z } p p

It may also happen that an interchange of two arguments causes the outcome to change sign.

Deﬁnition 2.16 Let sgn (π) denote the sign of the permutation π, i.e. sgn (π) = 1 if it is even, sgn (π) = −1 if 0 it is odd. T ∈ Tp(V ) is called an antisymmetric covariant tensor if for all v1,..., vp ∈ V and any permutation π = {π(1), . . . , π(p)} of labels {1, . . . , p},

T(vπ(1),..., vπ(p)) = sgn (π)T(v1,..., vp) .

p ∗ Likewise, T ∈ T0(V ) is called an antisymmetric contravariant tensor if for all vˆ1,..., vˆp ∈V

T(vˆπ(1),..., vˆπ(p)) = sgn (π)T(vˆ1,..., vˆp) .

Notation

V 0 • The vector space of antisymmetric covariant p-tensors is denoted p(V ) ⊂ Tp(V ). Vp p • The vector space of antisymmetric contravariant p-tensors is denoted (V ) ⊂ T0(V ). V ∗ ∗ Vp • Alternatively one sometimes writes p(V ) = V ⊗A ... ⊗A V and (V ) = V ⊗A ... ⊗A V . | {z } | {z } q p

It is convenient to admit tensor ranks p < 2 in this notation by a natural extension.

Assumption

W W0 V V0 • 0(V ) = (V ) = 0(V ) = (V ) = R. (This is consistent with Deﬁnitions 2.15 and 2.16 if we set π =id, sgn π =1 on the empty index set ∅).

W V ∗ • 1(V ) = 1(V ) = V . (This follows from Deﬁnitions 2.15 and 2.16, with π =id, sgn π =1 on {1}). • W1(V ) = V1(V ) = V . (This follows from Deﬁnitions 2.15 and 2.16, with π =id, sgn π =1 on {1}). 24 Tensor Calculus

2.8.3. Symmetry and Antisymmetry Preserving Product Operators

Ansatz

• An n-dimensional vector space V over R.

• A basis {ei} of V . W V Wp Vp • Tensor spaces of type p(V ), p(V ), (V ), (V ).

The outer product, recall Deﬁnition 2.13 on page 19 and in particular Theorem 2.4 on page 21, does not preserve (anti)symmetry. For this reason alternative product operators are introduced speciﬁcally designed for W (V ), V p p(V ), and their contravariant counterparts.

Deﬁnition 2.17 For every vˆ1,..., vˆk ∈ V ∗ we deﬁne the antisymmetric covariant k-tensor vˆ1 ∧ ... ∧ vˆk ∈ V k(V ) as follows: 1 k i vˆ ∧ ... ∧ vˆ (x1,..., xk) = dethvˆ , xji for all xj ∈ V .

Example Let vˆ, wˆ ∈ V ∗, x, y ∈ V , then

hvˆ, xi hvˆ, yi (vˆ ∧ wˆ )(x, y) = det = hvˆ, xihwˆ , yi − hwˆ , xihvˆ, yi . hwˆ , xi hwˆ , yi

Alternatively, vˆ ∧ wˆ = vˆ ⊗ wˆ − wˆ ⊗ vˆ . A more general result will be given on page 28.

Terminology The inﬁx operator ∧ is referred to as the wedge product or the antisymmetric product.

Observation

1 k i • eˆ ∧ ... ∧ eˆ (e1,..., ek) = detheˆ , eji = 1.

• eˆi1 ∧ ... ∧ eˆik (e ,..., e ) = δi1...ik , recall the completely antisymmetric symbols on page 5, cf. j1 jk j1...jk

• eˆi1 ⊗ ... ⊗ eˆik (e ,..., e ) = δi1 . . . δik . j1 jk j1 jk

Observation Furthermore, the wedge product is associative, bilinear and antisymmetric (whence nilpotent, vice versa), i.e. if uˆ, vˆ, wˆ ∈ V ∗, λ, µ ∈ R, then • (uˆ ∧ vˆ) ∧ wˆ = uˆ ∧ (vˆ ∧ wˆ ). • (λuˆ + µvˆ) ∧ wˆ = λuˆ ∧ wˆ + µvˆ ∧ wˆ . • uˆ ∧ (λvˆ + µwˆ ) = λuˆ ∧ vˆ + µuˆ ∧ wˆ .

• uˆ ∧ uˆ = 0, or, equivalently, uˆ ∧ vˆ = −vˆ ∧ uˆ.

A similar symmetry preserving product exists for symmetric tensors. To this end we need to introduce the symmetric counterpart of a determinant. 2.8 Tensors 25

Deﬁnition 2.18 The operator perm assigns a number to a square matrix in a way similar to det, but with a + sign for each term, recall page 4:

n n X 1 X perm A = |[j , . . . , j ]| A ...A = |[i , . . . , i ][j , . . . , j ]| A ...A . 1 n 1j1 njn n! 1 n 1 n i1j1 injn j1,...,jn=1 i1, . . . , in = 1 j1, . . . , jn = 1

Remark The formal role of the completely antisymmetric symbols in the above expressions is to restrict summation to relevant terms. The absolute value discards minus signs for odd parity terms, recall the last line in the observation made on page 4.

Terminology The operator perm is called the permanent.

Example Analogous to Table 1.1 on page 4 we have for a 3 × 3 matrix A,

perm A = A11A22A33 +A12A23A31 +A13A21A32 +A13A22A31 +A11A23A32 +A12A21A33 .

Deﬁnition 2.19 For every k-tuple vˆ1,..., vˆk ∈ V ∗ we deﬁne the symmetric covariant k-tensor vˆ1 ∨ ... ∨ vˆk ∈ W k(V ) as follows:

1 k i vˆ ∨ ... ∨ vˆ (x1,..., xk) = perm hvˆ , xji .

Example Let vˆ, wˆ ∈ V ∗, x, y ∈ V , then

hvˆ, xi hvˆ, yi (vˆ ∨ wˆ )(x, y) = perm = hvˆ, xihwˆ , yi + hwˆ , xihvˆ, yi . hwˆ , xi hwˆ , yi

Alternatively, vˆ ∨ wˆ = vˆ ⊗ wˆ + wˆ ⊗ vˆ .

Terminology The inﬁx operator ∨ is referred to as the vee product or the symmetric product.

Observation The vee product is associative, bilinear and symmetric, i.e. if uˆ, vˆ, wˆ ∈ V ∗, λ, µ ∈ R, then • (uˆ ∨ vˆ) ∨ wˆ = uˆ ∨ (vˆ ∨ wˆ ). • (λuˆ + µvˆ) ∨ wˆ = λuˆ ∨ wˆ + µvˆ ∨ wˆ . • uˆ ∨ (λvˆ + µwˆ ) = λuˆ ∨ vˆ + µuˆ ∨ wˆ .

• uˆ ∨ vˆ = vˆ ∨ uˆ. 26 Tensor Calculus

Observation

0 p • dim Tp(V ) = n .

i1 ip V •{ eˆ ∧ ... ∧ eˆ | 1 ≤ i1 < i2 < . . . < ip−1 < ip ≤ n} is a basis of p(V ).

i1 ip W •{ eˆ ∨ ... ∨ eˆ | 1 ≤ i1 ≤ i2 ≤ ... ≤ ip−1 ≤ ip ≤ n} is a basis of p(V ).

V W 0 • For p ≥ 2, dim p(V ) + dim p(V ) ≤ dim Tp(V ), with strict equality iff p = 2.

Caveat It is customary to suppress the inﬁx operator ∨ in the notation.

Theorem 2.5 Cf. the observations above. We have n n + p − 1 dim V (V ) = and dim W (V ) = . p p p p

Proof of Theorem 2.5.

• As for the first identity, a strictly increasing ordering of labels in a product of the form eî1 ∧ ... ∧ eîp can be realized in exactly as many distinct ways as one can select p items out of n different objects. V V • In particular, if p > n, then dim p(V ) = 0, i.e. p(V ) = {0}. • As for the second identity, the number of ways in which a non-decreasing ordering of labels in a product of the form eî1 ∨ ... ∨ eîp can be realized can be inferred from the following argument3. Any admissible ordering of labels can be represented symbolically by a sequence of p dots • , split into subsequences of equal labels separated by n − 1 crosses × , each of which marks a unit label transition. Examples, taking n = 5, p = 7: ••×•××•••×• ≡ eˆ1 ∨ eˆ1 ∨ eˆ2 ∨ eˆ4 ∨ eˆ4 ∨ eˆ4 ∨ eˆ5 ••×••••×•×× ≡ eˆ1 ∨ eˆ1 ∨ eˆ2 ∨ eˆ2 ∨ eˆ2 ∨ eˆ2 ∨ eˆ3 There are exactly (n + p − 1)! permutations of the n + p − 1 symbols in such sequences, with exactly p!, respectively (n−1)! indistinguishable permutations among the identical symbols • , respectively × , in any given realization.

Remark Notice the qualitative differences between V (V ) and W (V ) with respect to dimensionality scaling with p. In V p p the former case we have dim p(V ) = 0 as soon as p > n = dim V , i.e. no nontrivial antisymmetric tensors with ranks p exceeding vector space dimension n exist. In contrast, nontrivial symmetric tensors do exist for any rank. Example p Vp V 3 Let B and Bp denote bases for (V ), respectively p(V ), in Euclidean 3-space V = R .

p p B dim p Bp dim 0 {1} 1 0 {1} 1 1 2 3 1 {e1, e2, e3} 3 1 eˆ , eˆ , eˆ 3 1 2 1 3 2 3 2 {e1 ∧ e2, e1 ∧ e3, e2 ∧ e3} 3 2 eˆ ∧ eˆ , eˆ ∧ eˆ , eˆ ∧ eˆ 3 1 2 3 3 {e1 ∧ e2 ∧ e3} 1 3 eˆ ∧ eˆ ∧ eˆ 1

3Courtesy of Freek Paans. 2.8 Tensors 27

Observation Deﬁnitions 2.17, page 24, and 2.19, page 25, are straightforwardly complemented by similar ones applicable to (anti)symmetric contravariant tensors.

0 W Deﬁnition 2.20 The linear operator S : Tp(V ) → p(V ), given by 1 X (S (T)) (v1,..., vp) = T(v ,..., v ) , p! π(1) π(p) π in which summation runs over all permutations π of the index set {1, . . . , p}, is called the symmetrisation map.

Observation

i1 ip 0 i1 ip W • If T = ti1...ip eˆ ⊗ ... ⊗ eˆ ∈ Tp(V ), then S (T) = t(i1...ip)eˆ ⊗ ... ⊗ eˆ ∈ p(V ), in which

def 1 X t = t . (i1...ip) p! iπ(1)...iπ(p) π

W • If T = p(V ) is symmetric, then T = S (T). • The symmetrisation map is a projection, i.e. it is idempotent: S ◦ S = S .

1 1 X These observations explain the normalization factor , since 1 = 1. p! p! π

Observation

• The ∨-product is particularly convenient in the context of the symmetrisation operator, for we have 1 S (eî1 ⊗ ... ⊗ eîp ) = eî1 ∨ ... ∨ eîp . p!

W W • For T ∈ p(V ) and U ∈ q(V ), T ∨ U = U ∨ T. W • The case p = 0, q > 0 is covered by deﬁning λ ∨ U = λU for all λ ∈ R, U ∈ q(V ).

• Rule of thumb: The action of p!S on a ⊗-product amounts to a formal replacement ⊗ → ⊗S ≡ ∨.

The following result generalizes the example given on page 25:

(p + q)! Result 2.5 Let T ∈ W (V ) and U ∈ W (V ), then T ∨ U = S (T ⊗ U). p q p!q!

i1 ip ip+1 ip+q Proof of Result 2.5. Let T = ti1...ip eˆ ⊗ ... ⊗ eˆ and U = uip+1...ip+q eˆ ⊗ ... ⊗ eˆ , with symmetric

holors ti1...ip and uip+1...ip+q . Then

∗ 1 T ∨ U = t u (eî1 ⊗ ... ⊗ eîp ) ∨ (eîp+1 ⊗ ... ⊗ eîp+q ) = t u eî1 ∨ ... ∨ eîp+q i1...ip ip+1...ip+q p!q! i1...ip ip+1...ip+q (p + q)! (p + q)! ∗ i1 ip+q = ti ...i ui ...i S eˆ ⊗ ... ⊗ eˆ = S (T ⊗ U) . p!q! 1 p p+1 p+q p!q! In step ∗ we have used one of the observations on page 27 concerning the relationship between vee product and symmetrised outer product. The last step exploits linearity of S . 28 Tensor Calculus

0 V Deﬁnition 2.21 The linear operator A : Tp(V ) → p(V ), given by

1 X (A (T)) (v1,..., vp) = sgn (π)T(v ,..., v ) , p! π(1) π(p) π is referred to as the antisymmetrisation map.

Observation

i1 ip 0 i1 ip V • If T = ti1...ip eˆ ⊗ ... ⊗ eˆ ∈ Tp(V ), then A (T) = t[i1...ip]eˆ ⊗ ... ⊗ eˆ ∈ p(V ), in which

def 1 X t = sgn (π)t . [i1...ip] p! iπ(1)...iπ(p) π

V • If T = p(V ) is antisymmetric, then T = A (T). • The antisymmetrisation map is a projection, i.e. it is idempotent: A ◦ A = A .

Recall a previous remark on the normalization factor (page 27).

Observation

• The ∧-product is particularly convenient in the context of antisymmetrisation, for we have 1 A (eî1 ⊗ ... ⊗ eîp ) = eî1 ∧ ... ∧ eîp . p!

V • The case p = 0, q > 0 is covered by deﬁning λ ∧ U = λU for all λ ∈ R, U ∈ q(V ).

• Rule of thumb: The action of p!A on a ⊗-product amounts to a formal replacement ⊗ → ⊗A ≡ ∧.

The following result generalizes the example given on page 24.

(p + q)! Result 2.6 Let T ∈ V (V ) and U ∈ V (V ), then T ∧ U = A (T ⊗ U). p q p!q!

Notation

• The Einstein summation convention, recall page 7, is usually adapted in the context of antisymmetric V tensors. In general, if T ∈ p(V ), then we may use either of the following equivalent decompositions:

i1 ip X i1 ip – T = ti1...ip eˆ ⊗ ... ⊗ eˆ = ti1...ip eˆ ⊗ ... ⊗ eˆ ,

i1,...,ip

i1 ip X i1 ip – T = t|i1...ip|eˆ ∧ ... ∧ eˆ = ti1...ip eˆ ∧ ... ∧ eˆ . i1<...

Observation

i1 ip 1 i1 ip Note that t|i1...ip|eˆ ∧ ... ∧ eˆ = p! ti1...ip eˆ ∧ ... ∧ eˆ , recall the previous observation.

i1 ip ip+1 ip+q Proof of Result 2.6. Let T = t|i1...ip|eˆ ∧ ... ∧ eˆ and U = u|ip+1...ip+q |eˆ ∧ ... ∧ eˆ , with antisymmetric 2.8 Tensors 29

holors ti1...ip and uip+1...ip+q . Then

∗ 1 T ∧ U = t u eî1 ∧ ... ∧ eîp+q = t u eî1 ∧ ... ∧ eîp+q |i1...ip| |ip+1...ip+q | p!q! i1...ip ip+1...ip+q (p + q)! (p + q)! ? i1 ip+q = ti ...i ui ...i A eˆ ⊗ ... ⊗ eˆ = A (T ⊗ U) . p!q! 1 p p+1 p+q p!q!

In step ∗ we have used the fact that only distinct index values contribute nontrivially, together with the fact that there are precisely p! (q!) orderings of p (q) distinct indices. In step ? we have used the previous observation on the relationship between wedge product and antisymmetrised outer product. The last step exploits linearity of A .

V V pq Result 2.7 Let T ∈ p(V ) and U ∈ q(V ), then T ∧ U = (−1) U ∧ T.

Proof of Result 2.7. Cf. the proof of Result 2.6. To change the positions of the p basis covectors eî1 ,..., eîp in the product eî1 ∧ ... ∧ eîp+q so as to obtain the reordered product eîp+1 ∧ ... ∧ eîp+q ∧ eî1 ∧ ... ∧ eîp requires p (one for each eîk , k = 1, . . . , p) times q (one for each eî` , ` = p + 1, . . . , p + q) commutations, each of which brings in a minus sign.

Remark V V If p + q > n = dim V , then T ∧ U = 0 ∈ p+q(V ). Indeed, k(V ) = {0} for all k > n.

Remark

• Both S as well as A are idempotent: S ◦S = S , respectively A ◦A = A . Thus they are, by deﬁnition, 0 W V projections from Tp(V ) onto p(V ), respectively onto p(V ). • Note that we have not deﬁned the property of (anti)symmetry for a genuine mixed tensor.

• Most deﬁnitions also apply to the covariant and contravariant parts of a mixed tensor separately, i.e. treating the mixed tensor as a covariant tensor by freezing its covector arguments, respectively as a contravariant tensor by freezing its vector arguments. • A (say covariant) 2-tensor can always be decomposed into a sum of a symmetric and antisymmetric 2- 1 tensor, viz. T = S + A, with S = S (T) and A = A (T), i.e. S(v, w) = 2 (T(v, w) + T(w, v)) and 1 A(v, w) = 2 (T(v, w) − T(w, v)). This is consistent with the dimensionality equality for p = 2 in the observation on page 26. • A general tensor is typically neither symmetric nor antisymmetric, nor can it be split into a sum of a symmetric and antisymmetric tensor, recall the dimensionality inequality in the observation on page 26.

• The metric tensor G is symmetric, and so is its inverse, the dual metric tensor G−1. • Both the ∧-product as well as the ∨-product obey the “usual” rules for associativity, and distributivity with respect to tensor addition and scalar multiplication. 30 Tensor Calculus

Example 3 Let V = R , furnished with standard basis {e1, e2, e3}. One conventionally denotes ∂ ∂ ∂ e = e = e = . 1 ∂x 2 ∂y 3 ∂z

It is then customary to denote the corresponding dual basis {eˆ1, eˆ2, eˆ3} of V ∗ by

eˆ1 = dx eˆ2 = dy eˆ3 = dz .

i i i Let u = u ei, v = v ei, w = w ei ∈ V be arbitrary vectors. Then we have

dx(v) = hdx, vi = v1 , dy(v) = hdy, vi = v2 , dz(v) = hdz, vi = v3 .

Example Notation as in the previous example.

hdx, vi hdx, wi v1 w1 (dx ∧ dy)(v, w) = det = det = v1w2 − v2w1 , hdy, vi hdy, wi v2 w2 hdx, vi hdx, wi v1 w1 (dx ∧ dz)(v, w) = det = det = v1w3 − v3w1 , hdz, vi hdz, wi v3 w3 hdy, vi hdy, wi v2 w2 (dy ∧ dz)(v, w) = det = det = v2w3 − v3w2 . hdz, vi hdz, wi v3 w3

Example Notation as in the previous example.

 hdx, ui hdx, vi hdx, wi   u1 v1 w1  (dx ∧ dy ∧ dz)(u, v, w) = det  hdy, ui hdy, vi hdy, wi  = det  u2 v2 w2  hdz, ui hdz, vi hdz, wi u3 v3 w3 = u1v2w3 + u3v1w2 + u2v3w1 − u3v2w1 − u1v3w2 − u2v1w3 .

Notice that sofar no inner product has been invoked in any of the definitions and results of this section. With the help of an inner product certain “straightforward” modifications can be made that generalize foregoing definitions and results. The underlying mechanism is that of the conversion tools provided by Definition 2.9 on page 14.

Ansatz

• An n-dimensional inner product space V over R. W V Wp Vp • Tensor spaces of type p(V ), p(V ), (V ), (V ). 2.8 Tensors 31

Observation Recall Theorem 2.3, page 13, Deﬁnition 2.9 page 14, and Result 2.2, page 14. An inner product on V induces an Vp Wp inner product on the spaces (V ) and (V ), as follows. For any v1,..., vk, x1,..., xk ∈ V , set

• (v1 ∧ ... ∧ vk|x1 ∧ ... ∧ xk) = deth]vi, xji, respectively

• (v1 ∨ ... ∨ vk|x1 ∨ ... ∨ xk) = perm h]vi, xji. V W Likewise for the spaces p(V ) and p(V ): • vˆ1 ∧ ... ∧ vˆk|xˆ1 ∧ ... ∧ xˆk = dethvˆi,[xˆji, respectively

• vˆ1 ∨ ... ∨ vˆk|xˆ1 ∨ ... ∨ xˆk = perm hvˆi,[xˆji, for any vˆ1,..., vˆk, xˆ1,..., xˆk ∈ V ∗.

Section 2.8.5 covers tensors on an inner product space in more generality.

2.8.4. Vector Spaces with an Oriented Volume

Ansatz

• An n-dimensional vector space V over R.

• A preferred basis {ei} of V . V • A nonzero n-form µ ∈ n(V ).

Terminology

If µ(e1,..., en) = 1 then µ is called the unit volume form.

Observation

V • We have dim n(V ) = 1. V • Consequently, if µ1, µ2 ∈ n(V ), then µ1 and µ2 differ by a constant factor λ ∈ R. • For the unit volume form we have, according to the observation on page 28 (no P-convention after ∗),

n ∗ X 1 µ = eˆ1 ∧ ... ∧ eˆn = [i , . . . , i ] eî1 ∧ ... ∧ eîn = [i , . . . , i ] eî1 ∧ ... ∧ eîn , 1 n n! 1 n i1,...,in

which, by deﬁnition, can be expressed in terms of the general tensor basis, viz.

n n X i1 in ? X i1 in . i1 in µ = [i1, . . . , in] A eˆ ⊗ ... ⊗ eˆ = [i1, . . . , in] eˆ ⊗...⊗eˆ = µi1...in eˆ ⊗...⊗eˆ .

i1,...,in i1,...,in

Step ? exploits automatic antisymmetrization: The A -operator may be dropped, since its effect is void V when combined with antisymmetric coefﬁcients. Thus the covariant components of µ ∈ n(V ) are given

by µi1...in = [i1, . . . , in]. 32 Tensor Calculus

Caveat

The normalisation of the unit volume form µ requires a ﬁducial basis, viz. the preferred basis {ei} chosen above.

Consequently the identiﬁcation µi1...in = [i1, . . . , in] holds only relative to that basis! We refer to section 2.8.6 for a basis independent evaluation.

Observation

i1 in i1 in Thus µ = µ|i1...in|eˆ ∧ ... ∧ eˆ = µi1...in eˆ ⊗ ... ⊗ eˆ . This shows, once again, the convenience of the two variants of the Einstein summation convention.

V Deﬁnition 2.22 Let (V, µ) denote the vector space V endowed with the n-form µ ∈ n(V ), and let {ei} be a given basis of V .

• (V, µ) is called a vector space with an oriented volume.

• (V, µ) is said to have a positive orientation if µ(e1,..., en) > 0.

• (V, µ) is said to have a negative orientation if µ(e1,..., en) < 0.

Terminology The use of attributes “positive” and “negative” in this context is usually self-explanatory, e.g. V • given {ei} of V one refers to µ ∈ n(V ) as a positive volume form if µ(e1,..., en) > 0, and V • given µ ∈ n(V ) one refers to {ei} as a positively oriented basis if µ(e1,..., en) > 0, etc.

Notation

Instead of (V, µ) we shall simply write V . The basis {ei} relative to which µ(e1,..., en) = 1 holds should be clear from the context. Observation

Using Deﬁnition 2.17 on page 24 we may write, for x1,..., xn ∈ V ,

 1 1  heˆ , x1i ... heˆ , xni ˆj  . .  µ (x1,..., xn) = dethe , xii = det  . .  , n n heˆ , x1i ... heˆ , xni

which reveals the role of the Kronecker tensor (and thus of the dual space V ∗) in the deﬁnition of µ. Consistent i1 in with a foregoing observation it follows that the covariant components of µ = µi1...in eˆ ⊗ ... ⊗ eˆ are given j j by µi1...in = µ(ei1 ,..., ein ) = [i1, . . . , in] µ(e1,..., en) = [i1, . . . , in] detheˆ , eii = [i1, . . . , in] det δi = [i1, . . . , in] relative to (!) {ei}.

V Deﬁnition 2.23 Let a1,..., ak ∈V , k ∈ {0, . . . , n}, then we deﬁne the (n−k)-form µya1y ...yak ∈ n−k(V ) as follows:

(µya1y ...yak)(xk+1,..., xn) = µ (a1,..., ak, xk+1,..., xn) . 2.8 Tensors 33

Observation

• A trivial rewriting of the previous observation allows us to rewrite Deﬁnition 2.23 on page 32 as follows:

 1 1 1 1  heˆ , a1i ... heˆ , aki heˆ , xk+1i ... heˆ , xni  . . . .  (µya1y ...yak)(xk+1,..., xn) = det  . . . .  . n n n n heˆ , a1i ... heˆ , aki heˆ , xk+1i ... heˆ , xni

V • The holor of µya1y ...yak ∈ n−k(V ) for given a1,..., ak ∈ V is established as follows:

i1 ik (µya1y ...yak) eik+1 ,..., ein = a1 . . . ak (µyei1 y ...yeik ) eik+1 ,..., ein i1 ik i1 ik = a1 . . . ak µ (ei1 ,..., ein ) = µi1...in a1 . . . ak ,

ip in which we have used the decompositions ap = ap eip for p = 1, . . . , k together with multilinearity.

iq • By the same token, expanding also xq = xq eiq for q = k + 1, . . . , n, we obtain

i1 ik ik+1 in (µya1y ...yak)(xk+1,..., xn) = µi1...in a1 . . . ak xk+1 . . . xn .

Remark

V • An oriented volume µ ∈ n(V ) on a vector space V does not require an inner product. V ∗ • An oriented volume µ ∈ n(V ) exists by virtue of the existence of the dual space V . 1 n • If µ(e1,..., en) = 1, then µ = eˆ ∧ ... ∧ eˆ (unlike µ itself, the factors are not unique).

k+1 n • More generally, if µ(e1,..., en) = 1, then µye1y ...yek = eˆ ∧ ... ∧ eˆ . The following example explains the terminology of a “volume form”.

Example V Let the notation be as in the examples on page 30 and following. The space 3(V ) is spanned by {µ = dx ∧ dy ∧ dz}, which, according to the previous example, is given by

µ(u, v, w) = u1v2w3 + u3v1w2 + u2v3w1 − u3v2w1 − u1v3w2 − u2v1w3 .

This is the (signed) volume of a parallelepiped spanned by the (column vectors corresponding to the) vectors u, v, w. The result is positive (negative) if the ordered triple u, v, w has the same (respectively opposite) orientation as the basis {e1, e2, e3}.

The example suggests that similar interpretations hold for forms of lower degrees. This is indeed the case, as the next example shows. 34 Tensor Calculus

Example V 3 V 3 The spaces 1(R ) and 2(R ) are spanned by {dx, dy, dz} and {dx ∧ dy, dx ∧ dz, dy ∧ dz}, respectively. • According to the example on page 30 we have

hdx, vi = v1 hdy, vi = v2 hdz, vi = v3 .

One recognizes the (signed) 1-dimensional volumes (lengths) of the parallel projections of the vector v onto the x, y, respectively z axis. Recall the examples on page 17 for a physical interpretation in terms of particle kinematics or wave phenomena. • According to the example on page 30 we have

hdx, vi hdx, wi v1 w1 (dx ∧ dy)(v, w) = det = det = v1w2 − v2w1 . hdy, vi hdy, wi v2 w2

This is clearly the (signed) 2-dimensional volume (surface area) of the parallelogram spanned by v and w as it exposes itself from a perspective along the z direction, i.e. the area of the parallelogram after projection along the z axis onto the x-y plane, cf. Fig. 2.1.

Figure 2.1: Interpretation of (dx∧dy)(v, w) as the area of the cast shadow of a solar panel onto the ground (xy-plane) when lit from above (along z-direction). Note that (v∧w)(dx, dy) = (dx∧dy)(v, w) provides an equivalent interpretation of the same conﬁguration.

2.8.5. Tensors on an Inner Product Space

Ansatz An n-dimensional inner product space V over R. Heuristics The same mechanism that allows us to toggle between vectors and dual vectors in an inner product space, viz. via the ]- and [-operator associated with the metric G, also allows us to establish one-to-one relationships between p r Tq (V ) and Ts(V ), provided total rank is the same: p + q = r + s. This implies that—in an inner product space—tensor types can be ordered hierarchically according to their total ranks, and that, given ﬁxed total rank, a tensor can be represented in as many equivalent ways as there are ways to choose its covariant and contravariant rank consistent with its total rank. The basic principle is most easily understood by an example. 2.8 Tensors 35

Example 0 Let us interpret the metric G as an element of T2(V ), as previously discussed. We may consider the following four equivalent alternatives:

• G : V × V → R :(v, w) 7→ G(v, w);

def • H : V ∗ × V ∗ → R :(vˆ, wˆ ) 7→ H(vˆ, wˆ ) = G([vˆ,[wˆ );

def • I : V ∗ × V → R :(vˆ, w) 7→ I(vˆ, w) = G([vˆ, w);

def • J : V × V ∗ → R :(v, wˆ ) 7→ J(v, wˆ ) = G(v,[wˆ ). 0 2 1 1 Note that G ∈ T2(V ), H ∈ T0(V ), I ∈ T1(V ), J ∈ T1(V ). If {ei} is a basis of V , then i j i i • G = gijeˆ ⊗ eˆ = ]ei ⊗ eˆ = eˆ ⊗ ]ei;

ij i i • H = g ei ⊗ ej = [eˆ ⊗ ei = ei ⊗ [eˆ ;

i j i i • I = δjei ⊗ eˆ = ei ⊗ eˆ = [eˆ ⊗ ]ei;

j i i i • J = δi eˆ ⊗ ej = eˆ ⊗ ei = ]ei ⊗ [eˆ .

Caveat The above example declines from the strict ordering of the spaces V and V ∗ in the domain of deﬁnition of a tensor, as introduced in Deﬁnition 2.12 on page 18. This explains the existence of two distinct tensors, I 6= J, of 1 equal rank 1 . Note that I(vˆ, w) = J(w, vˆ).

Definition 2.24 Recall Result 2.2 on page 14 and Definition 2.12 on page 18. For any p, q ∈ N0 we define the 0 p covariant representation ]T ∈ Tp+q(V ) of T ∈ Tq (V ) as follows:

]T(v1,..., vp+q) = T(]v1,...,]vp, vp+1,..., vp+q) for all v1,..., vp+q ∈ V . p+q p Likewise, the contravariant representation [T ∈ T0 (V ) of T ∈ Tq (V ) is deﬁned as follows: ∗ [T(vˆ1,..., vˆp+q) = T(vˆ1,..., vˆp,[vˆp+1,...,[vˆp+q) for all vˆ1,..., vˆp+q ∈ V . p In general, a mixed representation of T ∈ Tq (V ) is obtained by toggling an arbitrary subset of (dual) vector spaces V and V ∗ in the domain of deﬁnition of the tensor by means of the metric induced {], [}-mechanism in a similar fashion.

Example 1 i j k i j Suppose T ∈ T2(V ) is given relative to a basis {ei} of V by T = t jkei ⊗ eˆ ⊗ eˆ . Let uˆ = uieˆ , vˆ = vjeˆ , k i j k i j k ijk wˆ = wkeˆ , u = u ei, v = v ej, w = w ek. Then ]T = tijkeˆ ⊗ eˆ ⊗ eˆ and [T = t ei ⊗ ej ⊗ ek. In terms of components: i j k T(uˆ, v, w) = t jkuiv w , i j k ]T(u, v, w) = T(]u, v, w) = tijku v w , (2.1) ijk [T(uˆ, vˆ, wˆ ) = T(uˆ,[vˆ,[wˆ ) = t uivjwk . The remaining ﬁve mixed representations are

ij k T1(uˆ, vˆ, w) = T(uˆ,[vˆ, w) = t kuivjw , i k j T2(uˆ, v, wˆ ) = T(uˆ, v,[wˆ ) = t j uiv wk , jk j T3(u, vˆ, wˆ ) = T(]u,[vˆ,[wˆ ) = ti uiv wk , (2.2) j i k T4(u, vˆ, w) = T(]u,[vˆ, w) = ti ku vjw , k i j T5(u, v, wˆ ) = T(]u, v,[wˆ ) = tij u v wk ,

making up the total of 23 = 8 different representations of the rank-3 tensor T, corresponding to the number of possibilities to toggle the arguments of the tensor’s 3 slots. 36 Tensor Calculus

Caveat The example shows that one quickly runs out of symbols if one insists on a unique naming convention for all distinct tensors that can be constructed in this way, especially for high ranks. One could opt for a systematic labeling, e.g. using a binary subscript to denote which argument requires a toggle (1) and which does not (0). In that case we would have, in the above example, T = T000, ]T = T100, [T = T011, T1 = T010, T2 = T001, T3 = T111, T4 = T110, T5 = T101.

Other labeling policies are possible, e.g. using a similar 3-digit binary code to denote argument type, 0 for a vector and 1 for a covector, say. This would lead to the following relabeling: T = T100, ]T = T000, [T = T111, T1 = T110, T2 = T101, T3 = T011, T4 = T010, T5 = T001.

Conventions like these are, despite the mathematical clarity they provide, rarely used. One often simply writes T for all cases, relying on the particular tensor prototype for disambiguation (“function overloading”).

Notation

• By abuse of notation one invariably uses the same symbol, T say, to denote any of the (covariant, contravariant, or mixed) prototypes discussed above.

S p • By the same token the space Tk(V ) = p+q=k Tq (V ) collectively refers to all tensors of total rank k. p • If a particular prototype T ∈ Tq (V ) is intended it should be clear from the context.

• Instead of the potentially confusing notation ti1...ir for a mixed holor one often writes ti1...ir , i.e. one j1...js j1...js with a manifest ordering of indices.

Observation

• The invocation of the ]-operator to a vector argument of the tensor affects the holor by an index lowering.

• The invocation of the [-operator to a covector argument of the tensor affects the holor by an index raising.

Caveat

j i −1 i ij • Recall that ]ei = G(ei) = gijeˆ , respectively [eˆ = G (eˆ ) = g ej. Thus, at the level of (dual) basis vectors, ] raises, and [ lowers indices! • Notice the way indices are raised or lowered in a mixed holor, viz. such as to respect the index ordering, cf. a previous remark on mixed holor notation.

Reminder

• In an inner product space, tensor (holor) indices can be raised or lowered with the help of the (dual) metric tensor. • As a rule of thumb, consistency of free index positioning determines whether and how to use metric and ij dual metric components, gij or g , in an expression involving ] or [.

2.8.6. Tensor Transformations 2.8 Tensors 37

2.8.6.1. “Absolute Tensors”

Ansatz

• An n-dimensional vector space V over R.

• A basis {ei} of V .

p • A linear space of mixed tensors, Tq (V ).

Recall that a holor depends on the chosen basis, but the corresponding tensor itself does not. This implies that holors transform in a particular way under a change of basis, which is characteristic for tensors. In fact, in the literature tensors are often deﬁned by virtue of the so-called tensor transformation law governing their holors. (Notice the misnomer.) Here we present it as a logical result of a tensor’s inherently basis independent nature.

i −1 Result 2.8 Let fj = Ajei deﬁne a change of basis, with transformation matrix A. Deﬁne B = A , i.e. AB = I, i k i or in terms of components, AkBj = δj. Then

i ...i def i ...i T = t 1 p e ⊗ ... ⊗ e ⊗ eˆj1 ⊗ ... ⊗ eˆjq = t 1 p f ⊗ ... ⊗ f ⊗ ˆf j1 ⊗ ... ⊗ ˆf jq , j1...jq i1 ip j1...jq i1 ip

iff the holor adheres to the tensor transformation law,

i ...i ` i k ...k t 1 p = A`1 ...A q Bi1 ...B p t 1 p . j1...jq j1 jq k1 kp `1...`q

ˆi i j Proof of Result 2.8. Suppose f = Cjeˆ . Then, by deﬁnition of duality,

i def î i k ` ∗ i ` k def i ` k i k δj = hf , fji = hCkeˆ ,Ajeì = CkAjheˆ , eì = CkAjδ` = CkAj ,

whence C = A−1 = B. Bilinearity of the Kronecker tensor has been used in ∗. Consequently,

` j j ˆk ei = Bi f` and eˆ = Akf .

Substitution into the basis expansion of T yields the stated result for the respective holors.

Remark The homogeneous nature of the tensor transformation law implies that a holor equation of the form ti1...ip = 0 j1...jq holds relative to any basis if it holds relative to a particular one.

Example i i With the same notation as used in Result 2.8 we have for a vector v = v ei = v fi:

i i j v = Bjv ,

also referred to as vector transformation law. Recall Theorem 2.1 on page 7. 38 Tensor Calculus

Example i ∗ 1 n Let df = ∂ifeˆ ∈ V denote the gradient of a scalar function f ∈ C (R ), interpreted as a dual (!) vector and i evaluated at a ﬁxed point. The interpretation of this is as follows. If v = v ei ∈ V , then

i ∗ hdf, vi = ∂ifv = (∇f|v) ,

the directional derivative of f along v. The equality marked by ∗ holds only if V is an inner product space, in which case we identify ∇f = [df ∈ V . ˆi ˆi i j According to the tensor transformation law we have df = ∂iff , with f = Bjeˆ , and

` ∂if = Ai ∂`f .

Notice the analogy with “inﬁnitesimal calculus”. Indeed, a change of coordinates, x 7→ x = φ(x), induces a i i reparametrization df = ∂ifdx = ∂ifdx , in which

∂f ∂f ∂x` ∂x` ∂ f = = = A`∂ f with A` def= , i ∂xi ∂x` ∂xi i ` i ∂xi by the chain rule.

Caveat One should distinguish between the new variable x and the coordinate transformation function φ that relates it to the original variable x by x = φ(x). In practice one often treats x either as a variable or as a function (i.e. x ≡ φ), depending on context. Likewise for x and φ−1.

Example Recall the observation on page 10, which conﬁrms the tensor transformation law for the dual basis vectors dxi:

∂xi dxi = dxj = Bidxj , ∂xj j with B = A−1, cf. the previous example.

Caveat i i j Compare the latter example of the transformation law for a basis covector, dx = Bjdx , to the one on page 37 i i j for the components (holor) of a vector, v = Bjv . The apparent formal correspondence has led practitioners of tensor calculus (mostly physicists and engineers) to interpret the basis covectors dxi as the “components of an infinitesimal displacement vector”. This misinterpretation is, although potentially confusing, harmless and indeed justified to the extent that it is consistent with the stipulated tensor transformation laws for basis dual vectors, respectively regular vector components. Confusion may be avoided altogether if we denote the components of an “infinitesimal vector” by δxi (i.e. a number, replacing the notation vi above so as to express its small nature), reserving dxi for the i-th dual basis covector (i.e. an element of V ∗).

2.8.6.2. “Relative Tensors”

i1 in Recall the unit volume form µ = µi1...in eˆ ⊗ ... ⊗ eˆ introduced in section 2.8.4. Recall from page 32 that the identiﬁcation µi1...in = [i1, . . . , in] depends on the preferred basis used for normalisation. Heuristically this is clear from the geometric interpretation provided by the example on page 33 in terms of the volume of the parallelepiped spanned by the ﬁducial basis vectors. Let us scrutinise the situation more carefully. 2.8 Tensors 39

Ansatz

• An n-dimensional vector space V over R.

• A ﬁxed basis {ei} of V .

i1 in i1 in V • The volume form µ = µ|i1...in|eˆ ∧ ... ∧ eˆ = µi1...in eˆ ⊗ ... ⊗ eˆ ∈ n(V ).

• The normalisation µi1...in = [i1, . . . , in], declaring µ as the unit volume form relative to {ei}.

The following lemma allows us to compute the holor of µ relative to an arbitrary basis, given its canonical form [i1, . . . , in] relative to our ﬁducial basis, via Result 2.8, page 37.

i1 in V Lemma 2.1 Let µ = µi1...in eˆ ⊗ ... ⊗ eˆ ∈ n(V ), with µi1...in = [i1, . . . , in] relative to basis {ei} of V , i î i ˆj î1 în and fj = Ajei, so that e = Ajf . Setting µ = µi1...in f ⊗ ... ⊗ f , we have

µ = Ai1 ...Ain µ = det A µ . j1...jn j1 jn i1...in j1...jn

Proof of Lemma 2.1. In accordance with Result 2.8, page 37, we have

µ = µ eˆi1 ⊗ ... ⊗ eˆin = µ Ai1 ...Ain ˆf j1 ⊗ ... ⊗ ˆf jn . i1...in i1...in j1 jn

This proves the ﬁrst identity. Using µi1...in = [i1, . . . , in], together with an observation made on page 5 in the context of the determinant, this may be rewritten as

ˆj1 ˆjn µ = det A [j1, . . . , jn] f ⊗ ... ⊗ f .

This establishes the proof of the last identity.

Observation

• Note that det A is the volume scaling factor of the transformation with matrix A.

• If µ (e1,..., en) = 1, then µ (f1,..., fn) = det A.

• Thus µ is, in general, not the unit volume form relative to {fi}, unless det A = 1. • The holor µ = [i , . . . , i ] is, consequently, not invariant under the absolute tensor transformation i1...in 1 n . law, Result 2.8, in the sense that µi1...in = det A µi1...in 6= [i1, . . . , in] = µi1...in .

Heuristics Clearly, the constant antisymmetric symbol fails to define an invariant holor of a tensor (as opposed to the holor i δj of the Kronecker tensor). The definition of the holor in terms of the antisymmetric symbol requires us to single out a preferred basis relative to which it holds, i.e. the fiducial basis {ei} has a preferred status among all possible bases. Lack of invariance is not surprising, since a given holor is, by its very definition, defined relative to a particular basis. The question nevertheless arises whether it is possible to give an invariant expression for the holor, i.e. one that holds in any basis, so that all bases are treated on equal foot and no a priori knowledge of a preferred one is needed. This is indeed possible by a formal “trick”, viz. through modification of the “tensor transformation law” for the (invariant counterpart of the) holor. There are two ways to remedy the lack of invariance in that case: 1. without the help of an inner product (Result 2.9), respectively 2. with the help of an inner product (Definition 2.25 and Result 2.10). 40 Tensor Calculus

Result 2.9 Recall Result 2.8, page 37, for notation. If

î1 în î1 în µ = µ|i1...in|e ∧ ... ∧ e = µ|i1...in|f ∧ ... ∧ f , and, by definition,

µ def= (det A)−1 Aj1 ...Ajn µ , i1...in i1 in j1...jn

then µi1...in ≡ µi1...in ≡ [i1, . . . , in] is an invariant holor.

Terminology

• This adapted transformation law is also known as the relative tensor transformation law. The attribute “relative” refers to the occurrence of (a power of) the Jacobian determinant. • By virtue of the relative tensor transformation law governing its holor, µ is referred to as a relative tensor. Strictly speaking this terminology refers to its holor deﬁnition!

Proof of Result 2.9. Using an observation on page 5 we ﬁnd

µ def= (det A)−1 Aj1 ...Ajn µ = (det A)−1 Aj1 ...Ajn [j , . . . , j ] i1...in i1 in j1...jn i1 in 1 n −1 = (det A) [i1, . . . , in] det A = [i1, . . . , in] = µi1...in .

Remark

• In Lemma 2.1 on page 39 the holor µi1...in = [i1, . . . , in] is transformed according to the absolute tensor

transformation law, recall Result 2.8 on page 37, which implies that µi1...in = det A µi1...in 6= µi1...in .

• In Result 2.9 the holor µi1...in = [i1, . . . , in] is transformed according to the relative tensor transformation

law, rendering the holor µi1...in = [i1, . . . , in] invariant in the sense that µi1...in = µi1...in = [i1, . . . , in] regardless of the basis.

• Adopting the relative tensor transformation law allows us to interpret µi1...in as a ‘covariant alias’ of the permutation symbol [i1, . . . , in], which by definition does not depend on a basis. • There is no inconsistency other than abuse of notation (to be resolved below). In both interpretations we have î1 în î1 în µ = µi1...in e ⊗ ... ⊗ e = µi1...in f ⊗ ... ⊗ f , or, equivalently, î1 în î1 în µ = µ|i1...in|e ∧ ... ∧ e = µ|i1...in|f ∧ ... ∧ f .

Notation The following convention is used henceforth to support consistent use of the Einstein summation convention and to resolve ambiguity w.r.t. the meaning of µi1...in :

i1...in • Henceforth we will adopt µi1...in and µ as covariant and contravariant aliases for the permutation i1...in symbol, i.e. µi1...in ≡ µ ≡ [i1, . . . , in].

• Consequently, the holor µi1...in is subject to the relative tensor transformation law, Result 2.9. 2.8 Tensors 41

2.8.6.3. “Pseudo Tensors”

Ansatz

• An n-dimensional inner product space V over R, based on our default Deﬁnition 2.5.

• A ﬁxed basis {ei} of V .

i1 in i1 in V • The unit volume form µ = µ|i1...in|eˆ ∧ ... ∧ eˆ = µi1...in eˆ ⊗ ... ⊗ eˆ ∈ n(V ), with invariant holor

deﬁnition µi1...in = [i1, . . . , in].

Notation

For brevity one denotes the determinant of the Gram matrix, gij = (ei|ej), by g = det gij, recall page 5 concerning a remark on the use of indices, and Deﬁnition 2.8 on page 12.

Deﬁnition 2.25 The Levi-Civita tensor is the unique unit n-form of positive orientation: √ √ 1 n i1 in = g µ = g eˆ ∧ ... ∧ eˆ = |i1...in|eˆ ∧ ... ∧ eˆ .

Observation

√ √ • i1...in = g [i1, . . . , in] = g µi1...in .

i1 in i1 in • = i1...in eˆ ⊗ ... ⊗ eˆ = |i1...in|eˆ ∧ ... ∧ eˆ in terms of the appropriate summation conventions.

Theorem 2.6 The contravariant representation of the Levi-Civita tensor, [ ∈ Vn(V ), recall Deﬁnition 2.24 on |i1...in| page 35, is given by [ = ei1 ∧ ... ∧ ein , with holor representation 1 i1...in = √ [i , . . . , i ] . g 1 n

Proof of Theorem 2.6. It sufﬁces to prove the validity of the holor representation by index raising:

√ 1 i1...in = gi1j1 . . . ginjn = gi1j1 . . . ginjn g [j , . . . , j ] = √ [i , . . . , i ] . j1...jn 1 n g 1 n

i1j1 injn ij In the last step we have made use of the fact that g . . . g [j1, . . . , jn] = det g [i1, . . . , in], an observation ij made on page 5, together with the fact that det g = 1/ det gij.

Observation

1 1 • i1...in = √ [i , . . . , i ] = √ µi1...in relative to the ﬁducial basis {e }. g 1 n g i

i1...in |i1...in| • [ = ei1 ⊗ ... ⊗ ein = ei1 ∧ ... ∧ ein .

Result 2.10 Recall Result 2.8, page 37, for notation. Let

i1 in = |i1...in|eˆ ∧ ... ∧ eˆ 42 Tensor Calculus

√ be the basis n-form relative to basis {ei}, with i1...in = g [i1, . . . , in]. Moreover, let

ˆi1 ˆin = |i1...in|f ∧ ... ∧ f

j be the basis n-form relative to basis {fi}, with fi = Ai ej. Then the pseudotensor transformation law,

def= sgn (det A)Aj1 ...Ajn , i1...in i1 in j1...jn √ renders the holor form invariant, in the sense that i1...in = g [i1, . . . , in]. Note that = sgn (det A) .

Proof of Theorem 2.10. By virtue of the transformation properties of gij, recall Result 2.8 on page 37, one obtains

k ` gij = Ai Ajgk` , whence √ pg = |det A| g . By deﬁnition of the pseudotensor transformation law we have, using an observation on page 5, √ def= sgn (det A)Aj1 ...Ajn = sgn (det A) Aj1 ...Ajn g µ i1...in i1 in j1...jn i1 in j1...jn √ √ p = sgn (det A) det A g µi1...in = |det A| g µi1...in = g µi1...in .

Recall that µi1...in and µi1...in have been used here as aliases for the same (basis independent) permutation symbol [i1, . . . , in].

Remark

• The Levi-Civita tensor is referred to as a pseudotensor, because its form invariant holor (!) transforms according to the pseudotensor transformation law of Result 2.10, not the absolute tensor transformation law of Result 2.8. √ • Form invariance, achieved by virtue of the factor g, should not be confused with numerical invariance. √ √ Clearly g 6= g, whence i1...in 6= i1...in , unlike the formal identity µi1...in = µi1...in ≡ [i1, . . . , in]. • In the case of a non-positive inner product, recall Deﬁnition 2.7, slight adaptations may be needed, a typical √ one being the replacement of the g factor by p|g| if g is negative.

The following deﬁnition is based on an earlier observation, recall page 5.

Deﬁnition 2.26 The generalised Kronecker, or permutation symbol, is deﬁned as

 δi1 . . . δi1   j1 jk  +1 if (i1, . . . , ik) even permutation of (j1, . . . , jk) , i1...ik . . δ = det  . .  = −1 if (i1, . . . , ik) odd permutation of (j1, . . . , jk) , j1...jk  . .   δik . . . δik 0 otherwise . j1 jk

Observation

• i1...in = µ µi1...in = δi1...in . j1...jn j1...jn j1...jn

i ...i • `1...`ki1...in−k = µ µ`1...`ki1...in−k = k!δ 1 n−k . `1...`kj1...jn−k `1...`kj1...jn−k j1...jn−k 1 • For the matrix A of a mixed tensor A ∈ T1(V ) we have det Ai = δi1...in Aj1 ...Ajn . 1 j n! j1...jn i1 in 2.9 The Hodge Star Operator 43

2.8.7. Contractions

Ansatz

• An n-dimensional vector space V over R.

• A basis {ei} of V .

p • A linear space of mixed tensors, Tq (V ).

1 Deﬁnition 2.27 Let T ∈ T1(V ). The trace, tr T ∈ R, of T is deﬁned as follows: i tr T = T(eˆ , ei) .

Remark

p • The tr -operator is applicable to any mixed tensor T ∈ Tq (V ) provided p, q ≥ 1, viz. by arbitrarily ﬁxing p − 1 covector and q − 1 vector arguments.

p−1 • Without “frozen” arguments one may interpret tr T as a mixed tensor tr T ∈ Tq−1(V ). • The tr -operator lowers both covariant and contravariant ranks by one.

Caveat

p • In general, the notation tr T is ambiguous for T ∈ Tq (V ), unless T is symmetric with respect to both covector as well as vector arguments. (This is manifest if p = q = 1.) • One could decorate the tr -operator with disambiguating subscripts and superscripts so as to indicate the covector/vector argument pair it applies to. This is hardly ever done in the literature. Instead, ambiguities should be resolved from the context.

2.9. The Hodge Star Operator

Ansatz

• An n-dimensional inner product space V over R furnished with a basis {ei}. √ 1 n i1 in V • The Levi-Civita tensor = g eˆ ∧ ... ∧ eˆ = |i1...in|eˆ ∧ ... ∧ eˆ ∈ n(V ).

V V Deﬁnition 2.28 Let a1,..., ak ∈ V . The Hodge star operator ∗ : k(V ) → n−k(V ) is deﬁned as follows: ∗1 = for k = 0, ∗ (]a1 ∧ ... ∧ ]ak) = ya1y ...yak for k = 1, . . . , n, followed by linear extension.

∗ ∗ This deﬁnition could be rephrased by replacing ]a1,...,]ak ∈ V by aˆ1,..., aˆk ∈ V on the l.h.s. and a1,..., ak ∈ V by [aˆ1,...,[aˆk ∈ V on the r.h.s. A similar deﬁnition could be given for a contravariant 44 Tensor Calculus

Vk Vn−k counterpart ∗ : (V ) → (V ), in which case ]-conversion of the vector arguments a1,..., ak ∈ V is obsolete. Remark V V The Hodge star operator establishes an isomorphism between k(V ) and n−k(V ), recall the dimensionality argument on page 26: n n dim V (V ) = = = dim V (V ) . k k n − k n−k

Observation

ip • The holor of the tensor ∗ (]a1 ∧ ... ∧ ]ak) for ﬁxed ap = ap eip , with p = 1, . . . , k, is obtained as follows:

i1 ik (∗ (]a1 ∧ ... ∧ ]ak)) eik+1 ,..., ein = a1 . . . ak (∗ (]ei1 ∧ ... ∧ ]eik )) eik+1 ,..., ein

def i1 ik = a1 . . . ak (yei1 y ...yeik ) eik+1 ,..., ein i1 ik i1 ik = a1 . . . ak (ei1 ,..., ein ) = i1...in a1 . . . ak .

iq • By the same token, expanding also xq = xq eiq for q = k + 1, . . . , n, we obtain

i1 ik ik+1 in (∗ (]a1 ∧ ... ∧ ]ak)) (xk+1,..., xn) = i1...in a1 . . . ak xk+1 . . . xn .

Result 2.11 Alternatively, we may rewrite Deﬁnition 2.28 as follows:

∗1 = for k = 0, 1 k 1 k ∗ aˆ ∧ ... ∧ aˆ = y[aˆ y ...y[aˆ for k = 1, . . . , n.

p In this case we ﬁnd, setting aˆp = a eˆjp for p = 1, . . . , k, and using eˆj = gij]e , jp i

∗ aˆ1 ∧ ... ∧ aˆk e ,..., e = a1 . . . ak ∗ eˆj1 ∧ ... ∧ eˆjk e ,..., e ik+1 in j1 jk ik+1 in = gi1j1 . . . gikjk a1 . . . ak (∗ (]e ∧ ... ∧ ]e )) e ,..., e j1 jk i1 ik ik+1 in def = gi1j1 . . . gikjk a1 . . . ak ( e ... e ) e ,..., e j1 jk y i1 y y ik ik+1 in = gi1j1 . . . gikjk a1 . . . ak (e ,..., e ) j1 jk i1 in = gi1j1 . . . gikjk a1 . . . ak . i1...in j1 jk

Observation As a rule of thumb, in order to compute the holor of the Hodge star of a general antisymmetric cotensor of rank k, the holor of which has free indices i1, . . . , ik, say, raise all indices and contract with the antisymmetric V V i1...in -symbol. That is, if A ∈ k(V ), then the holor of ∗A ∈ n−k(V ) is given by

def 1 ∗A = (∗A) e ,..., e = gi1j1 . . . gikjk A . ik+1...in ik+1 in k! i1...in j1...jk In other words, ik+1 in ik+1 in ∗A = ∗Aik+1...in eˆ ⊗ ... ⊗ eˆ = ∗A|ik+1...in|eˆ ∧ ... ∧ eˆ . In fact it sufﬁces to remember that ∗ (]ei1 ∧ ... ∧ ]eik ) eik+1 ,..., ein = i1...in , by exploiting multilinearity. 2.9 The Hodge Star Operator 45

Remark

• Taking k = 0 in the above observation is consistent with ∗1 = . • Taking k = n shows that ∗ = ∗∗ 1 = 1. This result relies essentially on Deﬁnition 2.5. In the general case of Deﬁnition 2.7 you may verify that ∗ = ∗∗ 1 = sgn g, and that ∗∗ A = (−1)k(n−k) sgn g A for a V general k-form A ∈ k(V ). 46 Tensor Calculus 3. Differential Geometry

3.1. Euclidean Space: Cartesian and Curvilinear Coordinates

In this chapter we consider tensor ﬁelds, i.e. tensors, in the widest sense as introduced in Chapter 2, attached to all points of an open subset Ξ ⊂ Rn and considered as “smoothly varying” functions of those points. The space Rn represents n-dimensional Euclidean space, i.e. “classical” space endowed with a ﬂat, homogeneous and isotropic structure as we usually take for granted [8, 9, 10].

Terminology Recall Deﬁnition 2.5, page 11. The terminology “n-dimensional Euclidean vector space” is synonymous to “n-dimensional (real, positive deﬁnite) inner product space”.

Deﬁnition 3.1 An n-dimensional Euclidean space E is a metric space, i.e. an afﬁne space equipped with a distance function d : E × E → R, furnished with a mapping + : E × V → E, in which V is an n-dimensional Euclidean vector space, such that

• for all x, y ∈ E there exists a unique v ∈ V such that y = x + v, and d(x, y) = p(v|v); • for all x, y ∈ E and all v ∈ V d(x + v, y + v) = d(x, y); • for all x ∈ E and all u, v ∈ V (x + u) + v = x + (u + v).

Heuristics Roughly speaking, a Euclidean space is a Euclidean vector space V that has “forgotten” its origin, or for which any point may be taken as origin. To make this more precise, let us introduce a scalar t ∈ R, replace y by x(t) and x by x(0), and consider the parametrized (straight) path x(t) = x(0) + vt. Recall that + : E × V → E does not denote vector addition here. If we now arbitrarily designate an origin ω ∈ E, and reinterpret x(t), x(0) ∈ E, by abuse of notation, as the (unique) vectors that connect them to this fiducial origin, then we may reinterpret each term as a genuine vector, and the transport operator as vector addition, + : V ×V → V . It is thus legitimate to read x(t) = x(0) + vt as a linear combination in V , with the identification V = E as a set, regardless of the . choice of origin. Note that x¨(t) =v ˙(t) = 0—with v(t) =x ˙(t)—is the defining second order ordinary differential equation for the family of all such (straight) lines. These are the so-called geodesics of Euclidean geometry, to be generalized for curved manifolds later on.

Observation The distance function naturally arises from the inner product: d(x, y) = p(y − x|y − x), in which y −x denotes the vector v ∈ V such that y = x + v. According to the explanation above we may reinterpret x, y ∈ E as vectors x, y ∈ V after the choice of a global origin, which boils down to replacing the distance function prototype d : E × E → R by one deﬁned on V , viz. d : V × V → R for the difference vector (with the − : E × E → V operator reinterpreted as − : V × V → V in the usual way, referring to vector addition with the antivector of its second argument). 48 Differential Geometry

Observation Another important observation is that if x(t) = x(0) + vt and y(t) = y(0) + wt are two Euclidean geodesics, . then z(t) = y(t)−x(t) is generically linear in t (constant iff v = w), whence its second order derivative vanishes identically, z¨(t) = 0. This is the so-called geodesic deviation equation of Euclidean geometry. It relates to Euclid’s fifth postulate about Euclidean geometry [8]: “If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.” (You may want to sketch the situation and relate Euclid’s fifth postulate to the geodesic deviation vector z(t).)

Caveat Both E and V are usually written as Rn, implying the choice of a (global) basis. It should be clear from the context whether a Euclidean space or a Euclidean vector space is meant.

A point in x ∈ E ∼ Rn can be represented by its Cartesian coordinates xi ∈ R relative to a (globally deﬁned) i n Cartesian coordinate system or standard basis {ei}, i = 1, . . . , n, in the usual way: x = x ei. Here ei ∈ R is the coordinate vector with an entry 1 in i-th position and with remaining n − 1 entries equal to 0.

It is sometimes convenient to consider (curvilinear) coordinates on (an open subset of) Rn. The formal deﬁnition of coordinates, generally applicable to both Euclidean as well as curved spaces, is given in the next section.

3.2. Differentiable Manifolds

Differentiable manifolds, aka differential manifolds, are spaces that locally look Euclidean. The points of such manifolds can be labelled with the help of an atlas, i.e. a collection of charts that provide us with coordinates on open sets that cover the entire manifold.

Deﬁnition 3.2 An n-dimensional smooth differentiable manifold is a Hausdorff topological space M furnished n with a family of smooth diffeomorphisms φα :Ωα ⊂ M → φα(Ωα) ⊂ R , with the following properties:

•{ Ωα} is an open covering of M, i.e. Ωα is open and ∪αΩα = M.

If , then n are open sets, and −1 and • Ωα ∩ Ωβ 6= ∅ φα(Ωα ∩ Ωβ), φβ(Ωα ∩ Ωβ) ⊂ R φα ◦ φβ φβ (Ωα∩Ωβ ) −1 (thus) φβ ◦ φ are diffeomorphisms. α φα(Ωα∩Ωβ )

• The atlas {(Ωα, φα)} is maximal w.r.t. the previous two axioms.

The last axiom ensures that any chart we may come up with is tacitly assumed to be already contained in the atlas. . . To keep notation simple one often writes x = φα(p) and y = φβ(p) for the coordinates of a ﬁducial point −1 −1 p = φα (x) = φβ (y) relative to their corresponding charts. The maps

−1 n n −1 φβ ◦ φα : φα(Ωα ∩ Ωβ) ⊂ R → φβ(Ωα ∩ Ωβ) ⊂ R : x 7→ y = (φβ ◦ φα )(x) (3.1) −1 n n −1 φα ◦ φβ : φβ(Ωα ∩ Ωβ) ⊂ R → φα(Ωα ∩ Ωβ) ⊂ R : y 7→ x = (φα ◦ φβ )(y) (3.2)

−1 are called coordinate transformations. You may read φα as “the coordinates of . . . ”, and φα as “the point on the −1 manifold with coordinates . . . ”, both relative to the coordinate chart labelled by α. Hence y = (φβ ◦φα )(x) may be pronounced as “the coordinates y on ‘page’ β of the same physical point as the one referred to by coordinates x on ‘page’ α in the atlas”. The intersection Ωα ∩Ωβ ⊂ M is, loosely speaking, that part of the physical manifold that is ‘visible’ on both ‘pages’ α and β of the atlas. 3.3 Tangent Vectors 49

Notation By abuse of notation Eqs. (3.1–3.2) are often simpliﬁed as y = y(x), respectively x = x(y), with proper domains tacitly suppressed.

Caveat The thing to keep in mind is that coordinates x, y ∈ Rn are subjective in the sense that they can be rather arbitrarily chosen. Eqs. (3.1–3.2) ensure that, given their associated charts (φα, φβ), both refer to the same physical point −1 −1 p = φα (x) = φβ (y) ∈ M. The consequence of this observation is that, in all that follows, we must ensure that numerical statements expressed in terms of coordinates are invariant under coordinate transformations. Here lies the connection with Chapter 2.

−1 −1 Deﬁnition 3.3 The Jacobian of the coordinate transformation φβ ◦φα , respectively φα ◦φβ , is the n×n matrix of ﬁrst order partial derivatives

−1 i i −1 i i ∂(φβ ◦ φ ) (x) . ∂y ∂(φα ◦ φβ ) (y) . ∂x Si(x) = α = resp. T i(y) = = . j ∂xj ∂xj j ∂yj ∂yj

Observation The matrices S and T are regular, with S = T inv by virtue of the chain rule. In simpliﬁed notation we have

∂yi ∂xk Si T k = = δi . k j ∂xk ∂yj j

3.3. Tangent Vectors

In differential geometry, the tangent space takes over the role of the archetypical vector space V that lies at the core of all tensor calculus concepts introduced in Chapter 2. You may see this as a local copy of V attached to each point x ∈ M of the base manifold, with dim V = dim M = n. For reason of clarity we write TMx for a n local copy of V attached to x, or, by abuse of notation, TMx if x = φα(x) ∈ R in some given coordinate chart.

The tangent space is the vector space consisting of all tangent vectors at a ﬁducial point.

Deﬁnition 3.4 Let γ :(−ε, ε) → M : t 7→ γ(t) be a smooth parametrized curve, with γ(0) = p ∈ M, and f ∈ C∞(M) an arbitrary smooth scalar ﬁeld. Then the tangent vector of γ at p is the linear map given by

∞ d γ˙ (0) : C (M) → R : f 7→ (f ◦ γ) . dt t=0

−1 Smoothness of γ and f means that the numerically valued functions γ = φα ◦ γ and f = f ◦ φα are smooth on n the open interval (−ε, ε) ⊂ R, respectively on the open neighbourhood φα(Uα) ⊂ R with p ∈ Uα, in the sense i ∞ ∞ of standard calculus, i.e. γ ∈ C ((−ε, ε)) for each component i = 1, . . . , n, and f ∈ C (φα(Uα)).

Now recall the observation on page 8 concerning the abstract notation of basis vectors as partial derivative operators, initially justiﬁed by the formal consistency of their ‘coordinate transformation properties’ (chain rule) with basis transformations. On a differentiable manifold we can make this more concrete, because we now actually have something to take derivatives of.

Evaluating the expression in Deﬁnition 3.4 on a dummy scalar ﬁeld with the help of a coordinate chart such that 50 Differential Geometry

n x = φα(x) ∈ φα(Uα) ⊂ R , we observe that

d d −1 d i ∂ . i ∂ γ˙ (0)f = (f ◦ γ) = f ◦ φα ◦ φα ◦ γ = f ◦ γ =x ˙ (0) i f =x ˙ (0) i f , dt t=0 dt t=0 dt t=0 ∂x x(0) ∂x γ(0) (3.3) . i . in which γi(t) = xi(t) and γ˙ (t) =x ˙ i(t). Here we have made use of the aforementioned coordinate prototypes

n . γ :(−ε, ε) ⊂ R → R : t 7→ x(t) = γ(t) , (3.4) n . −1 f : φα(Uα) ⊂ R → R : x 7→ f(x) = (f ◦ φα )(x) , (3.5)

and of the convenient deﬁnition ∂ . ∂ i f = i f . (3.6) ∂x x(0) ∂x γ(0) The dummy nature of f on left and right hand sides of Eq. (3.3) justiﬁes the notation introduced without derivative connotation on page 8.

Notation Recall Deﬁnition 3.4 and Eqs. (3.3–3.6). Instead of γ˙ (0) we write

i ∂ v|p = v i (3.7) ∂x p

i i in which v|p =γ ˙ (0) and v =x ˙ (0). The subscript is reminiscent of the fact that the tangent vector is deﬁned at p = γ(0). This construction can be repeated for any point x ∈ M and any smooth curve γ ∈ C∞((−ε, ε), M).

Observation Omitting explicit reference to the ﬁducial point p for the sake of simplicity, we observe from Eq. (3.3) that

vf = df(v) = hdf, vi .

In particular, if we take the coordinate scalar ﬁeld f i : M → R : x 7→ xi and the coordinate basis vector i i i i i vj = ∂j ∈ TMx in an arbitrary coordinate chart x = φα(x), then vjf = ∂jx = δj = dx (∂j) = hdx , ∂ji, which shows that, indeed consistent with the formal notation introduced on page 8, the 1-forms dxi are just the dual basis covectors for the coordinate vectors ∂j.

Result 3.1 Recall Section 2.3. Relative to a coordinate basis we have, at any implicit point p ∈ M,

i i hdx , ∂ji = δj .

Remark ∗ The base point p ∈ M of interest is important. It is ususally taken for granted that h , i ∈ T Mp ⊗ TMp, i ∗ dx ∈ T Mp, ∂j ∈ TMp, et cetera, without explicit reference to basepoint p ∈ M for simplicity of notation. If one wishes to be explicit p may be attached as a label to each local geometric entity as a reminder.

3.4. Tangent and Cotangent Bundle

Ansatz An n-dimensional manifold M.

Deﬁnition 3.5 The tangent space at x ∈ M, TMx, is the vector space of tangent vectors vx at point x ∈ M. 3.5 Exterior Derivative 51

Remark

Recall from the previous section that the deﬁnition can be stated in intrinsic terms by deﬁning TMx as the span of n tangent vectors of a family of mutually transversal curves intersecting at x ∈ M. A special case is obtained by considering the n local coordinate curves induced by the coordinate system at x ∈ M.

Terminology In the latter case the basis is referred to as a coordinate basis or holonomic basis, in which case the i-th coordinate basis vector ei is also often written as ∂i, recall the observation on page 8.

Deﬁnition 3.6 The tangent bundle, TM = ∪x∈MTMx, is the union of tangent spaces over all x ∈ M.

Observation

• In order to identify base points explicitly, one often incorporates the projection map, π : TM → M : (x, v) 7→ x, into the deﬁnition of the tangent bundle. In particular π(TMx) = x and π(TM) = M.

Notation

∗ • Recall Deﬁnition 3.5. The cotangent space at x ∈ M is denoted by T Mx.

∗ ∗ • Recall Deﬁnition 3.6. The cotangent bundle is written as T M = ∪x∈MT Mx.

3.5. Exterior Derivative

Ansatz The tangent bundle TM over an n-dimensional manifold M.

k i1 ik 1 i1 ik V Deﬁnition 3.7 Let ω = ω|i1...ik| dx ∧...∧dx = k! ωi1...ik dx ∧...∧dx ∈ k(TM) be an antisymmetric covariant tensor ﬁeld of rank k, or k-form for brevity, with x ∈ M. The exterior derivative of ω is the (k +1)-form given by 1 . 1 dωk = dω ∧ dxi1 ∧ ... ∧ dxik = ∂ ω dxi ∧ dxi1 ∧ ... ∧ dxik ∈ V (TM) , k! i1...ik k! i i1...ik k+1

def i with df = ∂if dx for any sufﬁciently smooth scalar ﬁeld f : M → R.

Remark k The factor 1/k! has been provided so that we may interpret ωi1...ik as the holor of ω , since 1 k i1 ik i1 ik i1 ik i1 ik ω = ω dx ∧...∧dx = ωi ...i dx ∧...∧dx = ωi ...i A (dx ⊗...⊗dx ) = ωi ...i dx ⊗...⊗dx , |i1...ik| k! 1 k 1 k 1 k in which the last step relies on automatic antisymmetrisation due to antisymmetry of the holor. Also note that

1 (k + 1)! ∂ ω dxi ∧ dxi1 ∧ ... ∧ dxik = ∂ ω dxi ∧ dxi1 ∧ ... ∧ dxik k! i i1...ik k! |i i1...ik|

Caveat We have loosely identified M ∼ Rn via an implicit choice of coordinates. Strictly speaking this can be done for sufficiently small neighbourhoods of any fiducial point x ∈ M and a corresponding open set of Rn. 52 Differential Geometry

Observation One readily veriﬁes the following properties:

n n V • dω = 0 for any ω ∈ n(TM).

2 k def k k V 2 • d ω = ddω = 0 for any ω ∈ k(TM) and k = 0, . . . , n, in other words: d = 0. • One sometimes writes d∧ instead of d.

Exterior derivatives are often used in combination with the Hodge star to extract derivatives from a k-form ﬁeld in some desired format. Note that d increases the rank of a k-form by 1, whereas ∗ converts a k-form into an (n−k)-form. The following example illustrates this up to ﬁrst order derivatives.

Example ∗ ∞ In the following examples, α = α1dx + α2dy + α3dz ∈ T M, with αi ∈ C (M) and dim M = 3. For simplicity the manifold is assumed to be Euclidean, and identiﬁed with R3 in Cartesian coordinates (in which the inner product deﬁning Gram matrix equals the identity matrix). Using ∗1 = dx ∧ dy ∧ dz, ∗dx = dy ∧ dz, ∗dy = −dx ∧ dz, ∗dz = dx ∧ dy, ∗(dx ∧ dy) = dz, ∗(dx ∧ dz) = −dy, ∗(dy ∧ dz) = dx, ∗(dx ∧ dy ∧ dz) = 1, we obtain the following results:

•∗ α = α3dx ∧ dy − α2dx ∧ dz + α1dy ∧ dz;

• dα = (∂xα2 − ∂yα1) dx ∧ dy + (∂xα3 − ∂zα1) dx ∧ dz + (∂yα3 − ∂zα2) dy ∧ dz;

•∗ dα = (∂xα2 − ∂yα1) dz − (∂xα3 − ∂zα1) dy + (∂yα3 − ∂zα2) dx;

• d ∗ α = (∂xα1 + ∂yα2 + ∂zα3) dx ∧ dy ∧ dz;

•∗ d ∗ α = ∂xα1 + ∂yα2 + ∂zα3; • et cetera.

Special cases arise if α = df for some f ∈ C∞(M), a so-called “exact 1-form”.

3.6. Afﬁne Connection

Ansatz The tangent bundle TM over an n-dimensional manifold M.

Recall the observation on page 8 concerning the formal notation for vector decomposition relative to a coordinate i ∞ basis, v = v ∂i. Also recall from inﬁnitesimal calculus that for a scalar function f ∈ C (M) we may deﬁne the i ∗ differential df = ∂ifdx ∈ T M. This suggests that we interpret a vector as a directional derivative operator, as follows.

∞ i Deﬁnition 3.8 Let f ∈ C (M) be a scalar function, and v = v ∂i ∈ TM. Then

def def i vf = hdf, vi = v ∂if .

So, when acting on a scalar ﬁeld it is tacitly understood that we make the formal identication v = hd · , vi.

Definition 3.9 An affine connection on a manifold M is a mapping ∇ : TM × TM → TM :(v, w) 7→ ∇vw, also ∞ referred to as the covariant derivative of w with respect to v, with the following properties. Let f, f1, f2 ∈C (M), v, v1, v2, w, w1, w2 ∈TM, and λ, λ1, λ2 ∈R, then 3.6 Affine Connection 53

∞ 1. ∇f1v1+f2v2 w = f1∇v1 w + f2∇v2 w for all f1, f2 ∈ C (M),

2. ∇v (λ1w1 + λ2w2) = λ1∇vw1 + λ2∇vw2, for all λ1, λ2 ∈ R,

∞ 3. ∇v(fw) = f∇v w + ∇vf w for all f ∈ C (M). Here we have deﬁned ∇vf ≡ vf.

Remark If we leave out the directional vector argument we call ∇w the covariant differential of w, and identify ∇f = df.

Thus the covariant derivative of a local frame vector (i.e. basis vector of the local tangent space) is a vector within the same tangent space. Consequently it can be written as a linear combination of frame vectors.

k k Deﬁnition 3.10 The Christoffel symbols Γij are deﬁned by ∇ej ei = Γijek.

Caveat

1. Conventions for the ordering of the lower indices of the Christoffel symbols differ among literature sources. As a rule of thumb I put the differentiation index in second position. 2. The Christoffel symbols are not the components of a tensor, v.i.

To see how the Christoffel symbols transform, consider a vector and dual covector basis transformation:

k k ` m e = S` e resp. ej = Tj em .

The matrices T and S are the mutually inverse Jacobian matrices:

∂xi ∂xi T i = resp. Si = . j ∂xj j ∂xj Set k k Γij = e , ∇ej ei

` ` Substitute the transformations, apply first and third rule of Definition 3.9, ∇ej = Tj ∇e` , respectively ∇e` (Ti ek) = k ` m ∂`Ti ek + Ti ∇e` ek, use the definition of the Christoffel symbols, ∇e` ek = Γkèm, and finally the chain rule, ` ∂j = Tj ∂`, to obtain the following result:

Result 3.2 The components of the Christoffel symbols in coordinate system x are given relative to those in coordinate system x by k k m n ` ` Γij = S` Tj Ti Γnm + ∂jTi .

Thus the tensor transformation law, recall Result 2.8 on page 37, fails to hold due to the inhomogeneous term, which arises after any non-afﬁne coordinate transformation.

Remark k For any ﬁxed point on the manifold you can solve the system of equations Γij = 0 by slick choice of Jacobian matrix, i.e. by transforming to a suitable local coordinate system. However, in general it is not possible to achieve this globally, unless the manifold happens to be ﬂat (zero Riemann curvature, v.i.).

Using the above deﬁnitions one can easily prove the following. 54 Differential Geometry

i i Result 3.3 Let v = v ei, w = w ei ∈ TM be two vector ﬁelds. Then i j ∇vw = v Diw ej . j j j k Here we have deﬁned the components of the covariant derivative Diw ≡ ∂iw + Γkiw .

Observation

• The meaning of Di applied to a holor is rank-dependent; the above deﬁnition applies to a holor with only one, contravariant index.

j th k • Moreover, Diw does not only depend on the j component of w, but on all components w , k =1, . . . , n.

Definition 3.11 See Definition 3.9. The affine connection can be extended to arbitrary tensor fields by general- p r izing the product rule, viz. for any pair of tensor fields, T ∈ Tq (TM) and S ∈ Ts(TM), say, we require

∇v (S ⊗ T ) = ∇vS ⊗ T + S ⊗ ∇vT .

Observation Recall the linear trace operator from Definition 2.27. For a covector field zˆ ∈ T∗M and a vector field w ∈ TM we have hzˆ, wi = tr (zˆ ⊗ v). Observe that

∇vhzˆ, wi = ∇v(tr (zˆ ⊗ v)) = tr (∇v(zˆ ⊗ v)) = tr (∇vzˆ ⊗ v) + tr (zˆ ⊗ ∇vv) = h∇vzˆ, wi + hzˆ, ∇vwi ,

j j j implying the product rule Di(zjw ) = Dizj w + zjDi w .

i ...i Result 3.4 See Result 3.3. Let T = T 1 p e ⊗...⊗e ⊗eˆj1 ⊗...⊗eˆjq and v = vie be a tensor, respectively j1...jq i1 ip i vector ﬁeld on M, then i ...i ∇ T = viD T 1 p e ⊗ ... ⊗ e ⊗ eˆj1 ⊗ ... ⊗ eˆjq . v i j1...jq i1 ip Here we have deﬁned the components of the covariant derivative i ...i i ...i ki ...i i i ...i k i ...i i ...i D T 1 p ≡ ∂ T 1 p + Γi1 T 2 p + ... + Γ p T 1 p−1 − Γk T 1 p − ... − Γk T 1 p . i j1...jq i j1...jq ki j1...jq ki j1...jq j1i kj2...jq jq i j1...jq−1k

With the latter deﬁnition the covariant derivative operator Di satisﬁes the product rule for general holor products. Note that Di is holor-type dependent, involving a ‘Γ-correction term’ for each index.

Terminology

• With affine geometry one indicates the geometrical structure of a manifold endowed with an affine connection. Note that this does not require the existence of a metric. Without a Riemannian metric, M is referred to as an affine space or affine manifold.

• If a metric (induced by an inner product on each TMx) does exist we are in the realm of Riemannian geometry. Endowed with a Riemannian metric, M is called a Riemannian space or Riemannian manifold. We refer to section 3.9 for further details.

Assumption

• Recall Definition 2.8, page 12. It is assumed that, if TMx is an inner product space for every x ∈ M, local bases are chosen in such a way that the coefficients of the Gram matrix, gij(x) = (ei|ej)x, vary smoothly ∞ across the manifold: gij ∈ C (M). (This is guaranteed if the basis is holonomic.) • Similar assumptions will henceforth be implicit in the case of tensor fields in general. 3.7 Lie Derivative 55

Notation The x-dependency of tensor ﬁelds is tacitly understood and suppressed in the notation.

3.7. Lie Derivative

Ansatz The tangent bundle TM over an n-dimensional manifold M.

Deﬁnition 3.12 The Lie derivative of w ∈ V with respect to v ∈ V is deﬁned as

Lvw = [v, w] ,

in which the so-called Lie bracket is deﬁned by the commutator

i j i j [v, w] f = v(wf) − w(vf) = v ∂iw − w ∂iv ∂jf .

for any smooth function f.

Observation

i j i j i • Lvw is itself a vector, with components [v, w] = v ∂jw − w ∂jv .

• No connection is involved, since only derivations of R-valued functions are considered. • Each term separately in the subtraction does not constitute a vector (it fails to satisfy the covariance requirement), but the difference does. • The Lie-bracket is antisymmetric.

• The Lie-bracket satisﬁes the Jacobi identity: [u, [v, w]] + [w, [u, v]] + [v, [w, u]] = 0.

3.8. Torsion

Ansatz

• The tangent bundle TM over an n-dimensional manifold M. • An afﬁne connection.

Deﬁnition 3.13 Notation as before. The torsion tensor is given by

T(v, w) = ∇vw − ∇wv − [v, w] .

Remark The torsion tensor thus reveals the difference between a commutator of ordinary derivatives, as in the Lie derivative, and of covariant derivatives induced by the afﬁne connection. 56 Differential Geometry

Observation i j k j Recall that ∇vw = v (∂iw + w Γki)∂j, from which you derive the components of the torsion tensor relative to a coordinate basis: j k i j j j T(v, w) = Tkiv w ∂j with Tki = Γik − Γki .

Observation The torsion tensor is antisymmetric: T(v, w) = −T(w, v).

3.9. Levi-Civita Connection

Ansatz

• The tangent bundle TM over an n-dimensional manifold M. • A Riemannian metric g ∈ T∗M ⊗ T∗M.

An important result is that a Riemannian geometry induces a unique torsion-free connection.

Definition 3.14 Recall Definition 3.12, page 55 and Definition 3.13, page 55. A metric connection or Levi-Civita connection on a Riemannian manifold (M, g) satisfies the following properties:

• z (g(v, w)) = g(∇zv, w) + g(v, ∇zw) for all v, w, z ∈ TM. • T(v, w) = 0 for all v, w ∈ TM.

Observation

The ﬁrst requirement is tantamount to the product rule: ∇z (v · w) = ∇zv · w + v · ∇zw for all v, w, z ∈ TM. Alternatively, in terms of the ]-operator, ∇ ◦ ] = ] ◦ ∇, recall Deﬁnition 2.9 on page 14.

Terminology

• The product rule is variously phrased as “covariant constancy of metric”, “preservation of metric”, “metric compatibility”, et cetera, and can be condensely written as ∇zg = 0 for all z ∈ TM. • The second requirement is referred to as the requirement of “zero torsion”.

Theorem 3.1 Recall Deﬁnition 3.14. The Levi-Civita connection is uniquely given by the Christoffel symbols 1 Γk = gk` (∂ g + ∂ g − ∂ g ) . ij 2 i `j j `i ` ij

Proof of Theorem 3.1. The proof is based on (i) the general properties of an affine connection, recall Definition 3.9 on page 52, and (ii) the defining properties of the Levi-Civita connection, Definition 3.14, page 56.

Here are the details. Show that relative to a basis the ﬁrst condition of Deﬁnition 3.14 is equivalent to Dmgij = 0, then cyclicly permute the three free indices; subtract the two resulting expressions from the given one. The result is ` ` ` ` ` ` ∂mgij − ∂igjm − ∂jgmi = gi` Γjm − Γmj + g`j Γim − Γmi − g`m Γij + Γji . 3.10 Geodesics 57

Subsequently, according to Deﬁnition 3.13 on page 55 and the subsequent observation on page 56, the second condition of Deﬁnition 3.14 is equivalent to ` ` Γij = Γji .

1 km ik i Apply this to the previous equality. Contraction with − 2 g on both sides, using the fact that g gkj = δj, ﬁnally yields the stated result.

Observation A metric connection is symmetric: k k Γij = Γji . Recall that this is not necessarily the case for an afﬁne connection in general.

3.10. Geodesics

1 i Definition 3.15 The covariant differential ∇v ∈ T1(TM) of a vector field v = v ∂i ∈ TM is defined as

k ∇v = ∇∂k vdx .

Observation j j j j k i j i j j j k If we deﬁne ∇v = Dv ∂j, then Dv = dv + Γkiv dx = Div dx , with Div = ∂iv + Γkiv .

Remark p The deﬁnition can be extended to arbitrary tensor ﬁelds T ∈ Tq (TM):

i ...i ∇T = DT 1 p dxj1 ⊗ ... ⊗ dxjq ⊗ ∂ ⊗ ... ⊗ ∂ , j1...jq i1 ip

with DT i1...ip = D T i1...ip dxi, recall Result 3.4. j1...jq i j1...jq

Heuristics The conceptual advantage of a (covariant) differential over a (covariant) derivative is that the former is applicable to vectors known only along a curve, whereas the latter requires us to specify a vector ﬁeld on a full neighbourhood, as the following deﬁnition illustrates.

Deﬁnition 3.16 A geodesic is a curve with “covariantly constant tangent”, or a curve obtained by “parallel i transport” of its tangent vector at any ﬁducial point. In other words, if v(t) =x ˙ (t)∂i for t ∈ I = (t0, t1), then

∇v(t) = 0 for all t ∈ I.

Remark

• It is tacitly understood that ∂i ∈ TMx(t) indicates the coordinate basis vector at point x(t). • One often identiﬁes v(x(t)) with v(t). It should be clear from the context which of these prototypes is applicable.

Observation k k k i j In terms of components we have Dv (t)/dt = 0, or x¨ + Γijx˙ x˙ = 0. 58 Differential Geometry

Terminology The latter second order ordinary differential equation is known as the geodesic equation.

Heuristics

• A geodesic generalizes the concept of a straight line, and can in general be seen as the “straightest line” between any two sufﬁciently close points on the geodesic. • This should not be confused with the “shortest line”, for this would require the additional notion of a “length” or “distance” on the manifold (i.e. a Riemannian metric).

3.11. Curvature

Deﬁnition 3.17 For v, w ∈TM the (antisymmetric) curvature operator, R(v, w):TM→TM:u7→R(v, w)u, is deﬁned as

R(v, w) = [∇v, ∇w] − ∇[v,w] .

Observation ∗ Alternatively, in order to obtain a scalar valued mapping, we would need a covector field bz ∈ T M besides the three vector field arguments u, v, w ∈ TM involved in Definition 3.17. This naturally leads to the definition of a fourth order mixed tensor field.

∗ ∗ Deﬁnition 3.18 For u, v, w ∈ TM, bz ∈ T M, the Riemann curvature tensor Riemann ∈ T M⊗TM⊗TM⊗TM is deﬁned as Riemann(bz, u, v, w) = hbz , R(v, w)ui .

Observation

• One should ﬁrst verify that Deﬁnition 3.17 does not require any derivatives of u!

ρ σ µ • In order to ﬁnd the components of the Riemann curvature tensor, write bz = zρdx , u = u ∂σ, v = v ∂µ, ν ρ σ µ ν w = w ∂ν . Then, using previous result, you ﬁnd R(bz, u, v, w) = Rσµν zρu v w , in which

ρ ρ ρ ρ ρ λ ρ λ Rσµν = hdx , R(∂µ, ∂ν )∂σi = ∂µΓσν − ∂ν Γσµ + ΓλµΓσν − Γλν Γσµ .

ρ ρ • Instead of Rσµν the holor is often written as R σµν in order to stress that the ﬁrst slot of the tensor is intended for the covector argument.

• Note that R(∂µ, ∂ν ) = [∇∂µ , ∇∂ν ] − ∇[∂µ,∂ν ] = [∇∂µ , ∇∂ν ], i.e. the curvature operator and, a fortiori, the Riemann curvature tensor apparently capture the intrinsic non-commutativity of covariant derivatives. • Thus the ‘second order’ nature of the curvature operator pertains to the geometry of the manifold, not the differential structure of the vector ﬁeld it operates on. • In the case of a Riemannian manifold (with the induced Levi-Civita connection) this is clear from the ρ appearance of (up to) second order partial derivatives of the Gram matrix gαβ in Rσµν . 3.12 Push-Forward and Pull-Back 59

Deﬁnition 3.19 The Ricci curvature tensor and the Ricci curvature scalar are deﬁned as

µ ν Ric(v, w) = Rµν v w , µν R = g Rµν ,

ρ with v, w ∈ TM, in which the Ricci holor is deﬁned as Rµν = Rµρν .

3.12. Push-Forward and Pull-Back

Ansatz

• A smooth mapping φ : M → N : x 7→ y = φ(x) between manifolds.

• A smooth scalar field f : N → R : y 7→ z = f(y). • A smooth vector field v ∈ TM. • A smooth covector field ν ∈ T∗N.

Deﬁnition 3.20

• The pull-back φ∗f ∈ C∞(M) of f ∈ C∞(N) under φ is deﬁned such that φ∗f = f ◦ φ.

∗ • The push-forward φ∗v ∈ TN of v ∈ TM under φ is deﬁned such that φ∗v(f) = v(φ f).

∗ ∗ ∗ ∗ • The pull-back φ ν ∈ T M of ν ∈ T N under φ is deﬁned such that hφ ν, viM = hν, φ∗viN.

Terminology The terminology reﬂects the “direction” of the mappings involved between (tangent or cotangent bundles over) the respective manifolds, cf. the diagram (3.8).

Remark

• Note that the dimensions of the manifolds M and N need not be the same. Observation

µ a ∂ • By decomposing φ∗v = v S ∈ TN it follows that the components of the push-forward are given by µ ∂ya ∂ya the Jacobian matrix of the mapping: Sa = . (Note: µ = 1, . . . , m = dim M, a = 1, . . . , n = dim N.) µ ∂xµ ∗ a µ ∗ • By the same token, decomposing φ ν = νaSµdx ∈ T M, it follows that the components of the pull-back are given by the same Jacobian. 60 Differential Geometry

φ TM −−−→∗ TN   π π y y M −−−→ N (3.8) φ x x π π   φ∗ T∗M ←−−− T∗N

3.13. Examples 3.13 Examples 61

3.13.1. Polar Coordinates in the Euclidean Plane

Example Starting out from a Cartesian coordinate basis at an arbitrary point of the Euclidean plane E (i.e. Euclidean 2-space), 1 0 e = , e = , x 0 y 1 we obtain the local polar coordinate basis by the following transformation:

∂xj ∂xj e = ∂ = ∂ = e . i i ∂xi j ∂xi j Here we identify x = (x, y), x = (r, φ), with

x(r, φ) = r cos φ y(r, φ) = r sin φ .

Recall that the Jacobian matrix and its inverse are given by

∂x` ∂xk T ` = resp. Sk = , i ∂xi ` ∂x` or, in explicit matrix notation (upper/lower index = row/column index):  x  −y p 2 2 ` cos φ −r sin φ x + y Ti = =  y  , sin φ r cos φ  x  px2 + y2  x y  cos φ sin φ ! p p k  x2 + y2 x2 + y2  S` = sin φ cos φ = y x . −  −  r r x2 + y2 x2 + y2

Consequently, writing “r” and “φ” for index values 1 and 2, respectively, ex cos φ er = cos φ sin φ = cos φ ex + sin φ ey = , ey sin φ ex −r sin φ eφ = −r sin φ r cos φ = −r sin φ ex + r cos φ ey = . ey r cos φ

Note that the polar basis depends on the base point of the tangent plane, whereas the Cartesian basis is identical in form at all base points. 62 Differential Geometry

Example

Cartesian coordinates x = (x, y) in the Euclidean plane E induce a local basis for each TEx relative to which ` all Γnm vanish identically (`, m, n = 1,..., 2). In terms of polar coordinates, x = (r, φ), we ﬁnd, according to Result 3.2 on page 53: k k ` Γij = S` ∂jTi . From the previous example it follows that for any (row) index k = 1, 2 and (column) index i = 1, 2,

cos φ sin φ ! 0 0 ! k ∂ cos φ −r sin φ Γ = sin φ cos φ = 1 , ir − ∂r sin φ r cos φ 0 r r r cos φ sin φ ! 0 −r ! k ∂ cos φ −r sin φ Γ = sin φ cos φ = 1 . iφ − ∂φ sin φ r cos φ 0 r r r

In other words, φ 1 φ 1 r k Γ = , Γ = , Γ = −r , all other Γ = 0. φr r rφ r φφ ij Consequently, 1 1 ∇ e = e , ∇ e = e , ∇ e = −re , all other ∇ e = 0. r φ r φ φ r r φ φ φ r ej i This can be immediately conﬁrmed by inspection of the (partial derivatives of the) coordinates of the polar basis vectors relative to the standard (Cartesian) basis, recall the previous example. 3.13 Examples 63

3.13.2. A Helicoidal Extension of the Euclidean Plane

Example Consider two coordinatizations of 3-space, x = (x, y, φ), and x = (ξ, η, θ), related as follows1:    x(ξ, η, θ) = ξ cos θ + η sin θ  ξ(x, y, φ) = x cos φ − y sin φ y(ξ, η, θ) = −ξ sin θ + η cos θ or η(x, y, φ) = x sin φ + y cos φ  φ(ξ, η, θ) = θ ,  θ(x, y, φ) = φ .

The Jacobian matrices are given by   ` cos θ sin θ −ξ sin θ + η cos θ ` ∂x Ti = i =  − sin θ cos θ −ξ cos θ − η sin θ  , ∂x 0 0 1

respectively  cos φ − sin φ −x sin φ − y cos φ  ∂xk Sk = = sin φ cos φ x cos φ − y sin φ . ` ∂x`   0 0 1

k k m n ` ` Again, recalling Result 3.2 on page 53: Γij = S` Tj Ti Γnm + ∂jTi . There are different ways to proceed, ` depending on our assumptions about the Γnm-symbols in (x, y, φ)-space. Here we pursue the simplest option, ` ignoring the remark in footnote 1, and assume Γnm = 0, i.e. (x, y, φ) is identiﬁed with a Cartesian coordinate triple in Euclidean 3-space. In this case we may evaluate the Christoffel symbols as in the previous example:

 cos φ − sin φ −x sin φ − y cos φ   cos θ sin θ −ξ sin θ + η cos θ  k ∂ Γ = sin φ cos φ x cos φ − y sin φ − sin θ cos θ −ξ cos θ − η sin θ , iξ   ∂ξ   0 0 1 0 0 1  cos φ − sin φ −x sin φ − y cos φ   cos θ sin θ −ξ sin θ + η cos θ  k ∂ Γ = sin φ cos φ x cos φ − y sin φ − sin θ cos θ −ξ cos θ − η sin θ , iη   ∂η   0 0 1 0 0 1  cos φ − sin φ −x sin φ − y cos φ   cos θ sin θ −ξ sin θ + η cos θ  k ∂ Γ = sin φ cos φ x cos φ − y sin φ − sin θ cos θ −ξ cos θ − η sin θ . iθ   ∂θ   0 0 1 0 0 1

A tedious but straightforward computation then reveals that

 0 0 0   0 0 1   0 1 −ξ  k k k Γiξ =  0 0 −1  Γiη =  0 0 0  Γiθ =  −1 0 −η  , 0 0 0 0 0 0 0 0 0

k η ξ η ξ ξ η k k i.e. all Γij = 0, except Γθξ = −1 , Γθη = 1 , Γξθ = −1 , Γηθ = 1 , Γθθ = −ξ , Γθθ = −η. Note that Γij −Γji = 0, consistent with the fact that torsion vanishes identically by construction.

1We interpret this space as an extension of Euclidean 2-space E by attaching to each point a copy of the periodic circle S = {φ ∈ [0, 2π) mod 2π} as an independent dimension. The parametrization can then be seen as a “shift-twist” transformation, whereby the Cartesian basisvectors ex and ey of the underlying Euclidean 2-space E (at any ﬁducial point) are rotated in accordance to the periodic translation in eφ-direction (generating S). 64 Differential Geometry

Example In the spirit of the previous example we consider a 3-space M, with coordinates (x, y, φ). However, we now introduce the following Riemannian metric g ∈ T∗M ⊗ T∗M:

g(x, y, φ) = dx ⊗ dx + dy ⊗ dy + (x2 + y2)dφ ⊗ dφ .

Note that the metric is form invariant under the reparametrization of the previous example:

g(ξ, η, θ) = dξ ⊗ dξ + dη ⊗ dη + (ξ2 + η2)dθ ⊗ dθ .

The induced Levi-Civita connection in (x, y, φ)-coordinates (and therefore also in (ξ, η, θ)-coordinates) follows from Theorem 3.1, page 56. In matrix notation we have for the metric and its dual, respectively,

 1 0 0   1 0 0  −1 G =  0 1 0  and G =  0 1 0  . 0 0 x2 + y2 0 0 1/(x2 + y2)

The fact that metric and its dual are diagonal greatly simpliﬁes the computation, for we have that 1 Γk = gkk (∂ g + ∂ g − ∂ g ) (no summation over k). ij 2 i kj j ki k ij

Moreover, by virtue of the fact that the coefﬁcients gab are constants for a, b = 1, 2, the above matrix (with row index i and column index j, and k ﬁxed) will have a null 2 × 2 block in the upper left part:

 0 0 ∗  k Γij =  0 0 ∗  (row index i, column index j, k ﬁxed.) ∗ ∗ ∗

Finally, the (dual) metric coefﬁcients do not depend on φ. By direct computation, taking these simplifying remarks into account, we obtain

 0 0 0   0 0 0   0 0 x  1 Γx = 0 0 0 Γy = 0 0 0 Γφ = 0 0 y . ij   ij   ij x2 + y2   0 0 −x 0 0 −y x y 0

k k Note that Γij − Γji = 0, consistent with the fact that torsion always vanishes in the case of the Levi-Civita connection. 3.13 Examples 65

Example Rotational invariance in the x–y plane of the previous example suggests the use of cylinder coordinates x = (r, ψ, θ) instead of x = (x, y, φ):   x(r, ψ, θ) = r cos ψ y(r, ψ, θ) = r sin ψ  φ(r, ψ, θ) = θ . To this end we once again invoke Result 3.2, page 53:

k k m n ` ` Γij = S` Tj Ti Γnm + ∂jTi .

The Jacobian matrices are given by   ` cos ψ −r sin ψ 0 ` ∂x Ti = i =  sin ψ r cos ψ 0  , ∂x 0 0 1

respectively  x/(x2 + y2) y/(x2 + y2) 0  ∂xk Sk = = −y/(x2 + y2) x/(x2 + y2) 0 . ` ∂x`   0 0 1 66 Differential Geometry Bibliography

[1] R. Abraham, J. E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis, and Applications, volume 75 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1988. [2] D. Bao, S.-S. Chern, and Z. Shen. An Introduction to Riemann-Finsler Geometry, volume 2000 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. [3] M. Berger. A Panoramic View of Riemannian Geometry. Springer-Verlag, Berlin, 2003.

[4] R. L. Bishop and S. I. Goldberg. Tensor Analysis on Manifolds. Dover Publications, Inc., New York, 1980. Originally published by The Macmillan Company in 1968. [5] M. P. do Carmo. Differential Geometry of Curves and Surfaces. Mathematics: Theory & Applications. Prentice-Hall, Englewood Cliffs, New Jersey, 1976.

[6] M. P. do Carmo. Riemannian Geometry. Mathematics: Theory & Applications. Birkhauser,¨ Boston, second edition, 1993. [7] E. Cartan. Lec¸ons sur la Geom´ etrie´ des Espaces de Riemann. Gauthiers-Villars, Paris, second edition, 1963. [8] Sir Thomas L. Heath. The Thirteen Books of Euclid’s Elements, volume 1 (Books I and II). Dover Publica- tions, Inc., New York, second edition, 1956. [9] Sir Thomas L. Heath. The Thirteen Books of Euclid’s Elements, volume 2 (Books III–IX). Dover Publica- tions, Inc., New York, second edition, 1956. [10] Sir Thomas L. Heath. The Thirteen Books of Euclid’s Elements, volume 3 (Books X–XIII). Dover Publica- tions, Inc., New York, second edition, 1956.

[11] J. Jost. Riemannian Geometry and Geometric Analysis. Springer-Verlag, Berlin, fourth edition, 2005. [12] J. J. Koenderink. Solid Shape. MIT Press, Cambridge, 1990. [13] A. P. Lightman, W. H. Press, R. H. Price, and S. A. Teukolsky. Problem Book in Relativity and Gravitation. Princeton University Press, Princeton, 1975.

[14] D. Lovelock and H. Rund, editors. Tensors, Differential Forms, and Variational Principles. Dover Publica- tions, Inc., Mineola, N.Y., 1988. [15] C. W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation. Freeman, San Francisco, 1973. [16] H. Rund. The Hamilton-Jacobi Theory in the Calculus of Variations. Robert E. Krieger Publishing Com- pany, Huntington, N.Y., 1973. [17] M. Spivak. Calculus on Manifolds. W. A. Benjamin, New York, 1965. [18] M. Spivak. Differential Geometry, volume 1–5. Publish or Perish, Berkeley, 1975.