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Home , Affine connection, Bundle metric, Cartan connection, Christoffel symbols, Connection (mathematics), Connection form

CARTAN AND FORMS

Syafiq Johar syafi[email protected]

Contents

1 Connections and Riemannian Curvature 1 1.1 Affine Connection ...... 1 1.2 Levi-Civita Connection and Riemannian Curvature ...... 2 1.3 Riemannian Curvature ...... 3

2 Covariant External 4 2.1 ...... 4 2.2 ...... 5 2.3 Explicit Formula for E = TM ...... 5 2.4 Levi-Civita Connection and Riemannian Curvature Revisited ...... 6

3 An Application: Uniformisation Theorem via PDEs 6

1 Connections and Riemannian Curvature

Cartan formula is a way of generalising the Riemannian curvature for a Riemannian (M, g) of n to a differentiable manifold. We start by defining the and . We begin by defining the usual notion of connection and the Levi-Civita connection.

1.1 Affine Connection

Definition 1.1 (Affine connection). Let M be a differentiable manifold and E a over M. A connection or at a p ∈ M is a map D : Γ(E) → Γ(T ∗M ⊗ E) with the ∞ properties for any V,W ∈ TpM, σ, τ ∈ Γ(E) and f ∈ C (M), we have that DV σ ∈ Ep with the following properties:

1. D is tensorial over TpM i.e.

DfV +W σ = fDV σ + DW σ.

2. D is R-linear in Γ(E) i.e. DV (σ + τ) = DV σ + DV τ.

3. D satisfies the Leibniz rule over Γ(E) i.e. if f ∈ C∞(M), we have

DV (fσ) = df(V ) · σ + fDV σ.

Note that we did not require any condition of the metric g on the manifold, nor the on E in the abstract definition of a connection.

1 Remark 1.1. Given connections DE and DF on vector bundles E and F respectively, we can define a connection DE⊗F on the bundle of E ⊗ F by:

E⊗F E F DX (σ ⊗ µ) = (DX σ) ⊗ µ + σ ⊗ (DX µ).

∗ Similarly, a connection DE on vector bundle E induces a connection DE on E∗ by using the and connection on E. Explicitly, if σ ∈ E and s ∈ E∗, we have the following:

E∗ E X(s(σ)) = (DX s)(σ) + s(DX σ).

Concretely, any affine connection is fully defined by the values of Dσ ∈ Γ(T ∗M ⊗ E) for all σ ∈ E. n We can choose a {σi}i=1 of E and define the values of Dσi for all i = 1, . . . , n. Therefore, for a given vector bundle on M, there are infinitely many choices of connections on any given vector bundle E. And one of the special one in the study of Riemannian is the Levi-Civita connection.

1.2 Levi-Civita Connection and Riemannian Curvature

Next, we can endow the vector bundle E with a bundle metric which is a family of inner products on

the fibres of Ex for all x ∈ M which varies smoothly on x ∈ M. In tensorial notation, a metric is an element of the symmetric 2-tensor h·, ·i ∈ Γ(Sym2(E∗)) which is positive definite. If E is the TM, then this bundle metric is just the Riemannian metric g.

Definition 1.2 (Compatible connection). A connection D on a vector bundle E is compatible with the metric h·, ·i on E if for any X ∈ Γ(TM) and σ, µ ∈ Γ(E), we have:

DX hσ, µi = Xhσ, µi = hDX σ, µi + hσ, DX µi

Theorem 1.1 (Fundamental theorem of ). On each Riemannian manifold (M, g), there exists a unique connection ∇ on TM which is compatible with the metric (i.e. ∇g ≡ 0) and

torsion-free (i.e. ∇X Y − ∇Y X = [X,Y ] for any vector fields X and Y ). This connection is called the Levi-Civita connection.

For vector fields X,Y ∈ Γ(TM), in local coordinates {xi}, we can write them explicitly as:

m m X i X i X = X (x)∂xi and Y = Y (x)∂xi , i=1 i=1

i i where X (x) and Y (x) are smooth functions on M for all i = 1, 2, . . . , m. Thus, the expression for ∇X Y can be computed directly by the properties of a connection in Definition 1.1 as follows:

j ∇X Y = ∇ i (Y ∂x ) X ∂xi j = Xi∇ (Y j∂ ) ∂xi xj = (Xi∂ Y j)∂ + XiY j(∇ ∂ ). xi xj ∂xi xj In the above expression, all of the terms are defined, except for ∇ ∂ ∈ Γ(TM). This can actually ∂xi xj k be written in terms of a collection of objects called the Christoffel symbols Γij i.e. ∇ ∂ = Γk ∂ for all i, j, k ∈ {1, 2, . . . , n}. (1) ∂xi xj ij xk Therefore, a Levi-Civita connection is fully characterised by these symbols. In local coordinates, the Christoffel symbols can be computed explicitly in terms of the metric g using the torsion free and compatibility properties. It is given as follows: 1 Γk = gkm ∂ g + ∂ g − ∂ g  . (2) ij 2 xj mi xi mj xm ij

2 k k Note the symmetry between the indices i and j i.e. Γij = Γji. Therefore, the Christoffel symbols is m2(m+1) simply a collection of 2 functions (which are fixed by the metric g) on the manifold (M, g). And thus, the Levi-Civita connection is completely defined by this choice of functions. With this out of the way, we can define the second covariant derivative, which in turn will be used to define curvature of a metric with this unique connection. The second covariant derivative is simply the composition of two covariant on a of a vector bundle. This is defined via the product rule.

Definition 1.3 (Second covariant derivative). We define the second covariant derivative on a vector bundle E as the map D2 : Γ(E) → Γ(T ∗M ⊗ T ∗M ⊗ E) given by

2 2 DXY σ = (D σ)(X,Y ) = (DX Dσ)(Y ) = DX (DY σ) − D∇X Y σ where ∇ is the Levi-Civita connection. This expression is C∞(M)-linear over X and Y .

Unlike the second order derivative in , the second covariant derivatives generally do not commute. In other words, we get the following:

2 2 D σ(X,Y ) − D σ(Y,X) = DX (DY σ) − DY (DX σ) − D[X,Y ]σ.

The expression above measures the failure for the second covariant derivative to commute. We can easily show that this quantity is in fact a tensor. This quantity is called the curvature of the connection, which is a central object of study in Riemannian geometry.

1.3 Riemannian Curvature

Definition 1.4 (Curvature). The curvature of the connection D on the vector bundle E is the tensor R ∈ Γ(T ∗M ⊗ T ∗M ⊗ E∗ ⊗ E) given by, for any X,Y ∈ Γ(TM) and σ ∈ Γ(E):

R(X,Y )σ = D2σ(X,Y ) − D2σ(Y,X)

= DX (DY σ) − DY (DX σ) − D[X,Y ]σ ∈ Γ(E). (3)

Definition 1.5 (Riemann curvature). When D is the Levi-Civita connection ∇ of Riemannian metric g (and hence E is TM), the quantity R in Definition 1.4 is called the Riemann curvature. In local i n coordinates {x }i=1, we can write the Riemann curvature as: ∂ R = R ldxi ⊗ dxj ⊗ dxk ⊗ . ijk ∂xl

l In fact, the quantity Rijk can be calculated explicitly since we know all the Christoffel symbols:

l l l l m l m Rijk = ∂xj Γik − ∂xi Γjk + ΓjmΓik − ΓimΓjk.

We also have the , also denoted R, which is defined as:

R(X,Y,Z,W ) = g(R(X,Y )Z,W ), for all W, X, Y, Z ∈ Γ(TM). In coordinates, the tensor can be written as:

i j k l R = Rijkldx ⊗ dx ⊗ dx ⊗ dx

m where Rijkl = glmRijk .

3 2 Covariant External Derivative

Of course, in the previous section, the Levi-Civita connection and the Riemannian curvature can be written explicitly in local coordinates. However, we want to find a more general form of connections and curvature without the presence of a metric by using covariant external derivative.

The Riemannian curvature depends on the Levi-Civita connection for ∇X Y to make sense. In par- ticular, we need to know how to differentiate a tangent bundle by setting a connection on T ∗M using the definitions/formulas (1) and (2). However, without a metric, we need an analogue of how to define this quantity for a general differentiable manifold. Recall that a connection must satisfy the Leibniz rule:

D(fσ) = dfσ + fDσ, for any smooth function f on M and σ ∈ Γ(E). Note that the d is just the on a 0-form. We can extend this definition onto higher degree differential forms Ωk(M) for k = 1, . . . , n. In particular, we are interested for the case of k = 1 which enables us to define the idea of curvature with respect to a connection. But first, let us define the connection.

2.1 Connection Form

Definition 2.1 (Covariant external derivative). Let E be a tensor bundle on M. A covariant external derivative (or external connection) is a map sending E-valued k-forms to E-valued k + 1-forms by the following rule:

D :Ωk(M,E) = Γ(Λk(T ∗M) ⊗ E) → Ωk+1(M,E) w ⊗ e 7→ dw ⊗ e + (−1)kw ∧ De, where De ∈ Ω(M,E) =∼ Γ(T ∗M ⊗ E).

Of course, as we have seen in the previous section, all of the terms in a covariant derivative is defined except for the terms of the form De ∈ Γ(T ∗M ⊗ E). Given an external connection D on the vector bundle E as defined in Definition 2.1 i.e. D : Γ(E) → Γ(T ∗M ⊗ E), we want to find an expression of n this connection by defining how D acts on elements e ∈ E, or sufficiently, on the basis {ei}i=1 of E. We define the connection form as follows:

j ∗ Definition 2.2 (Connection forms). A connection form wi on M is an n × n of one forms T M such that for a basis {ei} for E, we have

n X j ∗ Dei = wi ⊗ ej ∈ Γ(T M ⊗ E). (4) j=1

Thus, for any X ∈ Γ(TM), we have the following expression for a connection:

n X j DX ei = wi (X)ej ∈ Γ(E). j=1

j The choice for the matrix of one forms ωi fully characterises the connection on E. Therefore, again, we have infinitely many different external connections on a given vector bundle E on M.

j i Remark 2.1. One property of the connection forms is that it is anti-symmetric i.e. wi + wj = 0.

4 By properties of connections in Definition 1.1, we can extend this using Leibniz rule and so Pn i that for α ∈ Γ(E), we can write α = i=1 α ei and thus:

n n n X i X k X j k Dα = D(α ei) = dα ⊗ ek + α wj ⊗ ek = dα + wα. (5) i=1 k=1 j,k=1

2.2 Curvature Form

We can next define the curvature on a vector bundle E as the second covariant exterior derivative of (5). Note that the exterior covariant derivative on 1-forms is just the external derivative. From above, we have that for any α ∈ Γ(E), we have Dα = dα + wα. By replacing E with T ∗M ⊗ E in the above and taking the second exterior derivative of this, we get D(Dα) = D(dα + wα) = d(dα + wα) + w(dα + wα) = ¨d2¨α + dw ⊗ α − w ∧ dα + w ∧ dα + (w ∧ w)α = (dw + w ∧ w)α. This quantity is called the curvature form, denoted Ω. Formally, we have the following definition: Definition 2.3 (Curvature form). A curvature form Ω is an E valued 2-form defined as Ω = D ◦ D : Γ(E) → Γ(V2 T ∗M ⊗ E) is a matrix of 2-forms n j j X j k Ωi = dwi + wk ∧ wi . (6) k=1 Theorem 2.1 (Cartan structure equations). The two Cartan structure equations are: i Pn k i 1. dw = k=1 w ∧ wk, j j Pn j k 2.Ω i = dwi + k=1 wk ∧ wi .

2.3 Explicit Formula for E = TM

In fact, we can explicitly find a formula for the connection forms for E = TM. First, we define the dual bundle and dual frames: Definition 2.4 (Dual bundle). Given a vector bundle E over M. A dual vector bundle E∗ is the bundle of linear maps Hom(E, R × M). In other words, for each x ∈ M, the fibre of E∗ at x is the of linear maps from the vector space E(x) to the trivial fibre R × {x}.

Definition 2.5 (Dual frame). Let {e1(x), e2(x), . . . , en(x)} ⊂ Γ(E) be an frame of a rank n vector bundle E. We define the dual frame to be an ordered collection of n smooth section of the dual bundle 1 2 n ∗ i i (or covectors) {w (x), w (x), . . . , w (x)} ∈ Γ(E ) such that at each x ∈ M, w (ej) = δj for all i, j ∈ {1, 2, . . . , n}. Remark 2.2. By using the rule in Remark 1.1 to define a connection on a dual space, we can deduce that for the dual frame {wi} we have n i X j i Dw = − w ⊗ wj. j=1 n Remark 2.3. Using the first Cartan structure equation and a basis {ei}i=1 for E = TM, we would get the explicit formula for the connection forms: 1 wk(e ) = (dwi(e , e ) + dwj(e , e ) − dwk(e , e )), (7) i j 2 j k i k i j i n n where {w }i=1 is the dual basis with respect to {ei}i=1 and d is the usual exterior derivative on the space of 1-forms.

5 2.4 Levi-Civita Connection and Riemannian Curvature Revisited

Now we have seen the general connection and curvature form, how do they relate to the Christoffel symbols and the Riemannian curvature in the first section?

Remark 2.4. Note that given a local coordinate {xi}, we can construct a local orthonormal frame by Gram-Schmidt operation.

Now suppose that {∂i} is the local orthonormal frame of the tangent bundle of an m dimensional Riemannian manifold (M, g). The Levi-Civita connection acts as:

n X k ∇X ∂i = wi (X)∂k k=1

k k Thus, comparing with the formula for Christoffel symbols, we get that wi (ej) = Γij. Furthermore, if we compare equation (7) with the formula for Christoffel symbols in (2), we note the similarity in both of these equations, with (7) being the more general one. How about the curvature form? The curvature form in equation (6) coincides with the Riemann curvature tensor once we introduce a Riemannian metric g on M and use the Levi-Civita connection. In other words, the Riemannian curvature is derived from the curvature form by specifying the connection i forms to be the Christoffel symbols. Indeed, Ωj is a valued 2-form i.e. it is anti-symmetric in its arguments. It can be shown that

Theorem 2.2. We have: j Ωi (X,Y )∂j = R(X,Y )∂i.

Proof. Recall the definition for R(X,Y )∂i and expand the expression:

2 2 R(X,Y )∂i = ∇XY ∂i − ∇YX ∂i

= ∇X (∇Y ∂i) − ∇∇X Y ∂i − ∇Y (∇X ∂i) + ∇∇Y X ∂i j j j j = ∇X (wi (Y )∂j) − wi (∇X Y )∂j − ∇Y (wi (X)∂j) + wi (∇Y X)∂j j j j j = (∇X wi )(Y )∂j + wi (Y )(∇X ∂j) − (∇Y wi )(X)∂j − wi (X)(∇Y ∂j) j j k j j k = (∇X wi )(Y )∂j + wi (Y )wj (X)∂k − (∇Y wi )(X)∂j − wi (X)wj (Y )∂k j k j = dwi (X,Y )∂j + (wj ∧ wi )(X,Y )∂k j = Ωi (X,Y )∂j.

∗ by using the fact that dw(X,Y ) = (∇X w)(Y )−(∇Y w)(X) for any 1-form w ∈ Γ(T M) and the definition for wedge product.

j Thus, finally, by taking the inner product of Ωi (X,Y )∂j with ∂j, since {∂i} is chosen to be an j j i orthonormal frame, we have Ωi (X,Y ) = g(R(X,Y )∂i, ∂j). Note also that Ωi = −Ωj by the antisymmetry j in wi and wedge products of 1-forms, confirming the anti-symmetry of the last two arguments in the Riemann curvature tensor.

3 An Application: Uniformisation Theorem via PDEs

Conformal class of metrics are metrics which are defined up to scale. Any local angles on the manifold M are the same for metrics within the same conformal class. However, other geometric properties, such as curvature and lengths, may differ.

6 Definition 3.1 (Conformal class of metrics). A class of conformal metrics is a class of metrics which are equivalent under the relation: h ∼ g iff h = ug for some positive smooth function u : M → R.

Suppose that (Σ, g) is a with Gauss curvature K. Consider all the meticsg ˜ in the conformal class of g i.e.g ˜ = e2f g for some smooth function f : M → R. The uniformisation theorem asserts:

Theorem 3.1 (Uniformisation theorem). Every simply connected Riemannian surface (M, g) is confor- mally equivalent to a surface with constant of either −1, 0 or 1.

One way of proving this is using complex analysis. However, a more PDE oriented approach can be

used. On the given manifold (M, g), let {e1, e2} be an orthonormal frame for the tangent bundle with 1 2 i dual forms {w , w }. The first Cartan structure equation and the antisymmetry of wj asserts that:

1 2 1 dw = w ∧ w2, 2 1 2 dw = w ∧ w1.

Furthermore, since the only non-vanishing term in the Riemannian curvature tensor for (M, g) is R1221 = j K, using the relation Ωi (X,Y ) = g(R(X,Y )∂i, ∂j) and Cartan’s second structural equation, we have:

2 1 2 dw1 = −Kw ∧ w . (8)

Now we want to construct the same for the metricg ˜. We note that an orthonormal dual frame for the metricg ˜ is given by {w˜1 = ef w1, w˜2 = ef w2}. Thus, by Cartan’s first structural equation, we have:

1 2 1 dw˜ =w ˜ ∧ w˜2. (9)

On the other hand, by the definition ofw ˜1, we have:

1 f 1 f 1 1 2 1 f 2 1 dw˜ = df e ∧ w + e dw = (e1(f)w + e2(f)w ) ∧ w˜ + e (w ∧ w2) 2 1 2 1 2 1 1 = e2(f)w ˜ ∧ w +w ˜ ∧ w2 =w ˜ ∧ (e2(f)w + w2). (10)

1 1 1 2 Equating like terms in equations (9) and (10), we getw ˜2 = e2(f)w + w2. Similarly, we getw ˜1 = 2 2 2 1 e1(f)w + w1. These look like good candidates for the expression ofw ˜1 andw ˜2. However, recall that 2 1 the connection forms must satisfy anti-symmetry i.e.w ˜1 +w ˜2 = 0. So, we amend the terms by adding a 0 term to the RHS of (10), and get:

2 2 1 2 w˜1 = (e1(f)w − e2(f)w ) + w1, 2 1 2 1 w˜1 = (e2(f)w − e1(f)w ) + w2.

which satisfy the required antisymmetry. Now, recall equation (8) that on the Riemannian surface (M, g˜), we have 2 ˜ 1 2 dw˜1 = −Kw˜ ∧ w˜ . 2 2 2 1 2 ˜ 1 2 ˜ 2f 1 2 But on the other hand, dw˜1 = (e1(f) + e2(f) − K)w ∧ w and Kw˜ ∧ w˜ = Ke w ∧ w . Equating like terms, we get the PDE: 2f −∆gf + K = Ke˜ . To prove the uniformisation theorem is is the same as proving the existence of solution of this PDE for K˜ = −1, 0, 1.

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