Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions Introduction to Differential Geometry Lecture 7 of 10: Riemannian manifolds Dominic Joyce, Oxford University September 2019 2019 Nairobi Workshop in Algebraic Geometry These slides available at http://people.maths.ox.ac.uk/∼joyce/ 1 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions Plan of talk: 7 Riemannian manifolds 7.1 Riemannian metrics 7.2 The Levi-Civita connection 7.3 The Riemann curvature tensor 7.4 Volume forms and integrating functions 2 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions 7. Riemannian manifolds 7.1. Riemannian metrics n In Euclidean geometry on R , by Pythagoras' Theorem the distance between two points x = (x1;:::; xn) and y = (y 1;:::; y n) is 1 1 2 n n 2 1=2 n dR (x; y) = (x − y ) + ··· + (x − y ) : Note that squares of distances, rather than distances, behave 1 n n nicely, algebraically. If γ = (γ ; : : : ; γ ) : [0; 1] ! R is a smooth n path in R , then the length of γ is Z 1hdγ1 2 dγn 2i1=2 l(γ) = + ··· + dt: 0 dt dt Note that this is unchanged under reparametrizations of [0; 1]: ~ dγi dt replacing t by an alternative coordinate t multiplies dt by dt~ and dt~ dt by dt , which cancel. 3 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions n Regarding γ : [0; 1] ! R as a smooth map of manifolds, we have Z 1 dγ dγ 1=2 n l(γ) = gR jγ(t) (t); (t) dt; 0 dt dt dγ n n ∼ n 1 2 n 2 where dt (t) 2 Tγ(t)R = R , and gR jx = (dx ) + ··· + (dx ) in n n n n 2 ∗ ∼ 2 ∗ n 1 2 ∗ S Tx R = S (R ) for x 2 R , so that gR 2 C (S T R ). This is a simple example of a Riemannian metric on a manifold, being used to define lengths of curves. 4 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions Definition Let X be a manifold. A Riemannian metric g (or just metric) is a 2 ∗ 2 ∗ smooth section of S T X such that gjx 2 S Tx X is a positive definite quadratic form on Tx X for all x 2 X . In index notation we write g = gab, with gab = gba. We call (X ; g) a Riemannian manifold. Let γ : [0; 1] ! X be a smooth map, considered as a curve in X . The length of γ is Z 1 dγ dγ 1=2 l(γ) = gjγ(t) (t); (t) dt: 0 dt dt If X is (path-)connected, we can define a metric dg on X , in the sense of metric spaces, by dg (x; y) = inf l(γ): γ:[0;1]!XC 1: γ(0)=x; γ(1)=y Roughly, dg (x; y) is the length of the shortest curve γ from x to y. 5 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions Restricting metrics to submanifolds Let i : X ! Y be an immersion or an embedding, so that X is a submanifold of Y , and g 2 C 1(S2T ∗Y ) ⊆ C 1(N2 T ∗Y ) be a Riemannian metric on Y . Pulling back gives i ](g) 2 C 1(S2i ∗(T ∗Y )). We have a vector bundle morphism di : TX ! i ∗(TY ) on X , which is injective as i is an immersion, and a dual surjective morphism (di)∗ : i ∗(T ∗Y ) ! T ∗X . Symmetrizing gives S2(di)∗ : S2i ∗(T ∗Y ) ! S2T ∗X . Define i ∗(g) = (S2(di)∗)(i ](g)) 2 C 1(S2T ∗X ). This is defined for any smooth map i : X ! Y . But if i is an immersion, so that di : TX ! i ∗(TY ) is injective, then g positive definite implies i ∗(g) positive definite, so i ∗(g) is a Riemannian metric on X . We call it the pullback or restriction of g to X , and write it as gjX . 6 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions Submanifolds of Euclidean space Example n n 1 2 n 2 1 2 ∗ Define gR = (dx ) + ··· + (dx ) in C (S (R ) ). This is a n Riemannian metric on R , which induces the usual notions of n lengths of curves and distance in Euclidean geometry. We call gR n the Euclidean metric on R . Example n n Let X be any submanifold of R . Then gR jX is a Riemannian metric on X . n Since any manifold X can be embedded in R for n 0 (Whitney Embedding Theorem), this implies Corollary Any manifold X admits a Riemannian metric. 7 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions These ideas are important even in really basic applied mathematics, physics, geography, etc: Example 2 Model the surface of the earth as a sphere SR of radius 3 R = 6; 371km about 0 in R . Then the Riemannian metric gE = g 3 j 2 determines lengths of paths on the earth. Define R SR spherical polar coordinates (θ; ') (latitude and longitude) on 2 SR nfN; Sg by x(θ; ')=(R sin θ cos '; R sin θ sin '; R cos θ). Then 1 2 2 2 3 2 gE = (dx ) + (dx ) + (dx ) jS2 = (d(R sin θ cos '))2 + (d(R sin θ sin '))2 + (d(R cos θ))2 = R2(cos θ cos ' dθ − sin θ sin ' d')2 + (cos θ sin ' dθ + sin θ cos ' d')2 + (− sin θ dθ)2 = R2[dθ2 + sin2 θ d'2]: 8 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions 7.2. The Levi-Civita connection Any Riemannian manifold (X ; g) has a natural connection r on TX , called the Levi-Civita connection. This is known as the `Fundamental Theorem of Riemannian Geometry'. Theorem (The Fundamental Theorem of Riemannian Geometry) Let (X ; g) be a Riemannian manifold. Then there exists a unique connection r on TX ; such that r is torsion-free, and the induced connection r0 on N2 T ∗X satisfies r0g = 0. We call r the Levi-Civita connection of g. Usually we write r for all the induced connections on Nk TX ⊗ Nl T ∗X , without comment. So r allows us to differentiate all tensors T on a Riemannian manifold (X ; g), without making any arbitrary choices. 9 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions Proof of the Fundamental Theorem The theorem is local in X , so it is enough to prove it in coordinates (x1;:::; xn) defined on open U ⊆ X . Let r be a connection on a TX , with Christoffel symbols Γbc : U ! R, as in x6.5, so Pn a b n g = a;b=1 gab dx ⊗ dx with gab = gba. Then gab a;b=1 is a symmetric, positive-definite, invertible matrix of functions on U. abn Write g a;b=1 for the inverse matrix of functions. Then r torsion-free is equivalent to a a Γbc = Γcb; (7.1) 0 and rc gab = 0 is equivalent to @ g − Γd g − Γd g = 0: (7.2) @xc ab ac db bc ad a Calculation shows that (7.1){(7.2) have a unique solution for Γbc : a 1 ad @ @ @ Γ = g gdb + gdc − gbc : (7.3) bc 2 @xc @xb @xd This gives the unique connection r we want. 10 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions 7.3. The Riemann curvature tensor Let (X ; g) be a Riemannian manifold. Then by the FTRG we have a natural connection r on TX . As in x6.3, the curvature of r is R 2 C 1(TX ⊗ T ∗X ⊗ Λ2T ∗X ), which is called the Riemann a curvature tensor of g. In index notation R = Rbcd , and it is characterized by the formula for all vector fields u; v; w 2 C 1(TX ) a b c d Rbcd u v w c d a c d a c d c d a = v rc (w rd u )−w rc (v rc u )−(v rc w −w rc v )rd u c d a a = v w (rc rd u − rd rc u ); (7.4) d c d c d using [v; w] = v rc w − w rc v as r is torsion-free. Thus a b a Rbcd u = (rc rd − rd rc )u : (7.5) 11 / 36 Dominic Joyce, Oxford University Lecture 7: Riemannian manifolds Riemannian metrics Riemannian manifolds The Levi-Civita connection More about Riemannian manifolds The Riemann curvature tensor Volume forms and integrating functions Riemann curvature in coordinates Let (X ; g) be a Riemannian manifold with Riemann curvature R, and (x1;:::; xn) be coordinates on U ⊆ X .