Math 6396 Riemannian Geometry, Metric, Connections, Curvature Tensors etc. By Min Ru, University of Houston 1 Riemannian Metric • A Riemannian metric on a differentiable manifold M is a symmetric, positive-definite, (smooth) (0, 2)-tensor fields g on M, i.e., for any vec- tor field X, Y ∈ Γ(TM), g(X, Y ) = g(Y, X), g(X, X) ≥ 0 where the equality holds if and only if X = 0. In terms of the local coordinates, g can be expressed by ! i j ∂ ∂ g = gijdx ⊗ dx , gij = g , , ∂xi ∂xj where gij = gji and the matrix (gij) is positive definite everywhere. • There exists a Riemannian metric on a differentiable manifold with countable basis (for the topology). 2 Affine Connections • A affine (or linear) connection 5 is a map 5 : Γ(TM) × Γ(TM) → Γ(TM) which satisfies, for all X, Y, Z ∈ Γ(TM) and f ∈ C∞(M), (a) 5X+fY Z = 5X Z + f 5Y Z, (b) 5X (Y + Z) = 5X Y + 5X Z, (c) 5X (fY ) = (Xf)Y + f 5X Y . 1 • Levi-Civita connection. Let (M, g) be a Riemannian manifold. Then there exists a unique connection (called the Levi-Civita connec- tion) such that (1) 5X Y − 5Y X = [X, Y ] (i.e. torsion free) (2) X < Y, Z >=< 5X Y, Z > + < Y, 5X Z > . Proof: We first make the following remark: Let α : Γ(TM) → C∞(M) be a C∞(M)-linear, regarding Γ(TM) as an C∞(M)-module. Then there exists a unique U ∈ Γ(TM) such that α(Z) =< U, Z > for every Z ∈ Γ(TM). Outline of the proof: First we can derive, from the conditions that the Koszul formula holds: 2 < 5X Y, Z > = X < Y, Z > +Y (X, Z > −Z < X, Y > − < Y, [X, Z] > − < X, [Y, Z] > − < Z, [Y, X] > . This shows the uniqueness. Also, from the above discussion, if we define 5X Y use Koszul formula, we can verify the properties are satisfied. • Christoffel symbols. In terms of the coordinate chart (U, xi), we write m ∂ X k ∂ 5 ∂ = Γji , ∂xi j k ∂x k=1 ∂x k where the functions Γji are called the Christoffel symbols. k k • The connection 5 is torsion free if and only if Γij = Γji, and if 5 is torsion free and compatible with the Riemannian metric g, then its Christoffel symbols are given by, from the Koszul formula above, ∂g ∂g ∂g 2Γk g = il + jl − ij , ij kl ∂xj ∂xi ∂xl or equivalently, m ! k 1 X kl ∂gil ∂gjl ∂gij Γij = g j + i − l . 2 l=1 ∂x ∂x ∂x 2 With coordinate charts (U, xi) and (V, yi). On U ∩ V , the Christoffel symbols satisfy the following transformation formula m j p r 2 p j 0j X q ∂y ∂x ∂x ∂ x ∂y Γ ik = Γpr q i k + i k · p . p,q,r=1 ∂x ∂y ∂y ∂y ∂y ∂x For every X, Y ∈ Γ(TM), write m X i ∂ X = ξ i , i=1 ∂x m X j ∂ Y = η j , j=1 ∂x then k X X i ∂η X k i j ∂ 5X Y = ξ i + Γjiξ η k . k i ∂x i,j ∂x • In terms of the coordinate chart (U, xi), we write m ∂ X k ∂ 5 ∂ = Γji , ∂xi j k ∂x k=1 ∂x or m m m ∂ X X k i ∂ X k ∂ 5 j = Γjidx ⊗ k = ωj k , ∂x i=1 k=1 ∂x k=1 ∂x where m k X k i ωj = Γjidx . i=1 k The matrix ω = (ωi ) is called the connection matrix of 5 with respect to ∂/∂xi, 1 ≤ i ≤ m. So we see that we can regard 5 as the operator 5 : Γ(TM) → Γ(T ∗(M) ⊗ T (M)). • Parallel translation, geodesics. Suppose γ(t), 0 ≤ t ≤ b, is a curve in M. A (tangent) vector field X ∈ Γ(TM) (actually X can be the vector which is only defined along γ) is said to be parallel along the 3 curve γ if and only if 5γ0(t)X = 0 for t ∈ [0, b]. In terms of the local coordinate (U, xi), write by xi(t) = xi ◦ γ(t), 1 ≤ i ≤ m and write m X i ∂ X(t) = X (t) i |γ(t). α=1 ∂x Then i 0 X dx (t) ∂ γ (t) = i . i dt ∂x Hence k j X dX (t) X k i dx (t) ∂ 0 5γ (t)X = + ΓijX (t) k |γ(t). k dt i,j dt ∂x Hence X(t) is parallel along the curve γ if and only if k j dX (t) X k i dx (t) + ΓijX (t) = 0. dt i,j dt A curve γ is geodesic in M if and only if the tangent vector of γ is parallel along the curve γ. Suppose that γ is given in a coordinate chart (U, xi) by xi(t) = xi ◦ γ(t) 1 ≤ i ≤ m. Then γ is geodesic if and only if 2 i m j k d x X i dx dx 2 + Γkj = 0 1 ≤ i ≤ m. dt j,k=1 dt dt • Covariant derivative(connection) for tensor fields. Note that for any (fixed) tangent vector field X, according to the definition above, the operator 5X sends (1, 0)-tensor field Y (i.e. tangent vector field) to (1, 0)-tensor field 5X Y . Here we want to show that such 5 induces, for any (fixed) tangent vector field X, a map (still denoted by 5X which sends (0, s)-tensor fields to (0, s)-tensor fields. We first consider s = 0 case. Let f be a function (i.e. a (0, 0)-tensor field, then define 5X (f) = X(f). When s = 1, for any (0, 1)-tensor field ω (which is a differential 1-form), 5X (ω) is also a 1-form which is defined by, for any Y ∈ Γ(T (M)), 5X (ω)(Y ) = 5X (ω(Y )) − ω(5X Y ). 4 In general, let Φ be a (0, s)-tensor field (resp. a (1, s)-tensor field), and let X ∈ Γ(TM). Then we define the Covariant derivative of Φ in the direction of X by the formula, for every Y1,...,Ys ∈ Γ(T (M)), (5X Φ)(Y1,...,Ys) : = 5X (Φ(Y1,...,Ys)) s X − Φ(Y1,...,Yi−1, 5X Yi,Yi+1,...,Ys). i=1 5X Φ is then also a (0, s)-tensor (resp. a (1, s)-tensor), and 5Φ is a (0, s + 1)-tensor (resp. a (1, s + 1)-tensor) by means of the formula 5Φ(X, Y1,...,Ys) := (5X Φ)(Y1,...,Ys). Let (U, xi) be a local coordinate. Let Φ be a (0, s)-tensor field. Write j1 js Φ|U = Φj1···js dx ⊗ · · · ⊗ dx , then one can verify that j1 js 5 ∂ Φ = Φj1···js,idx ⊗ · · · ⊗ dx , ∂xi where ∂Φ s Φ = j1···js − X Φ Γk . j1···js,i i j1···jb−1kjb+1···js jbi ∂x b=1 • Connections over vector bundles. Let π : E → M be a vector bundle. Denote by Γ(E) the set of smooth sections of E. Γ(E) is a real vector space; it is also a C∞(M)-module. A connection 5 over E is a map 5 : Γ(T (M) × Γ(E) → Γ(E), (X, s) 7→ 5X s, which satisfies, ∞ for all X, Y ∈ Γ(T (M)), s1, s2 ∈ Γ(E), c constant, and f ∈ C (M): (a) 5X+fY s = 5X s + f 5Y s, (b) 5X (s1 + cs2) = 5X s1 + c 5X s2, (c) 5X (fs) = (Xf)s + f 5X s. 5 Note that we can also view 5 as the map Γ(E) → Γ(T ∗(M) ⊗ E) by (5s)(X) = 5X s for every s ∈ Γ(M),X ∈ Γ(TM). Note that if E = TM, then 5 is the connection we introduced earlier. Let 5 be a connection on TM, according to the discussion earlier, it induces a connection on the cotangent bundle T ∗M defined by, for every ω ∈ Γ(T ∗M), (5X ω)(Y ) = X(ω(Y )) − ω(5X Y ), for every Y ∈ Γ(T (M)). There exists a connection on every vector bundle π : E → M if M is a differentiable manifold with countable basis (for the topology). Let π : E → M be a vector bundle of rank k. Let 5 be a connection on E, i.e. 5 : Γ(E) → Γ(T ∗(M) ⊗ E) Let (U, xi) be a local coordinate of M. Let {sα, 1 ≤ α ≤ k} be a basis for Γ(E|U ) (they are called the local frame field). Then, for every s ∈ Γ(E), write k X β 5sα = ωα ⊗ sβ. β=1 β So, the connection 5 is associated to a k × k matrix ω = (ωα) of smooth differential one forms on U, with respect to the local frame {sα, 1 ≤ α ≤ k}. It is easy to verify that m β X β k ωα = Γαkdx , k=1 j where Γik are the Christoffel symbols. The matrix ω is called the connection matrix of 5 with respect to {sα, 1 ≤ α ≤ k}. Let s ∈ Γ(E) and write k X α s = λ sα, α=1 Then k k X α X β α 5s = dλ + λ ωβ ⊗ sα. α=1 β=1 6 For X ∈ Γ(TM), k k X α X β α 5X s = Xλ + λ ωβ (X) ⊗ sα. α=1 β=1 • Curvature tensor. For every X, Y ∈ Γ(TM), define R(X, Y ) (called the curvature operator) as R(X, Y ) : Γ(TM) → Γ(TM) by, for every Z ∈ Γ(TM), R(X, Y )Z = 5X 5Y Z − 5Y 5X Z − 5[X,Y ]Z.