Riemannian Connections 1. General Definition of a Riemannian Connection. Let (M, g) be a smooth manifold with a smooth Riemannian metric g. Let V be the set of all smooth vector fields on M. A connection or a covariant derivative on M is an operator ∇ : V × V → V, that, using the notation ∇(X, Y ) = ∇X Y , satisfies the following axioms: (1) ∇X1+X2 Y = ∇X1 Y + ∇X2 Y (2) ∇X (Y1 + Y2) = ∇X Y1 + ∇X Y2 (3) ∇fX Y = f∇X Y , where f : M → R is any smooth function. (4) ∇X (fY ) = X(f)Y + f∇X Y , where f : M → R is any smooth function. A connection ∇ on M is called a Riemannian connection with respect to g if in addition to the above axioms, ∇ also satisfies the following two axioms: (5) X g(Y, Z) = g(∇X Y, Z) + g(Y, ∇X Z) for any X, Y, Z ∈ V. (6) ∇X Y − ∇Y X = [X, Y ]. Theorem. For any smooth manifold M with a smooth Riemann- ian metric g there exists a unique Riemannian connection ∇ on M corresponding to g. 2. The case of Rn. Let M = Rn with the standard inner product giving on Rn. Then ∇X Y := DX Y is a Riemannian connection on M. Here DX Y is the directional derivative of Y with respect to X: Y (p + tX) − Y (p) Y (c(t)) − Y (p) DX Y |p := lim = lim t→0 t t→0 t where c(t) is any smooth curve with c(0) = p andc ˙(0) = X(p). Let c(t) be a curve and Y (t) be a vector field along c, that is for every t Y (t) is a tangent vector to R3 at the point c(t). Then the directional derivative of Y with respect to c is: Y (t + s) − Y (t) Dc˙Y (t) = lim . s→0 s Note that Dc˙c˙ =c ¨. 3. Riemannian connection on a surface in R3. 1 2 Let M 2 ⊆ R3 be an oriented surface with an outward unit normal n. We endow M with the Riemannian metric g obtained by restricting the standard inner product in R3 to the tangent vectors to M. For any tangent fields X, Y to M in R3 define ∇X Y := DX Y − hDX Y, nin, where DX Y is the directional derivative of Y with respect to X. Then ∇ is a Reimannian connection on M [Check that all the axioms hold!]. Note that in the above definition ∇X Y is the tangential (with respect to M) component of the vector DX Y . That is, ∇X Y is obtained by taking the orthognal projection of the vector DX Y to the tangent space of M at the given point. In view of the uniqueness of Riemannian connections ∇ is in fact the only Riemannian connection on M. 4. Riemannian connection in a chart. Let (M n, g) be a smooth manifold with a smooth Riemannian metric g. Let ∇ be the unique Riemannian connection on M corresponding to g. Let (U, φ = (x1, . , xn)) be a chart on M. We will use the notation ∇iY := ∇ ∂ Y . ∂xi k (1) The Christophell symbols Γij and Γij,k are defined as follows: ∂ X ∂ ∇ = Γk , i ∂xj ij ∂xk k and ∂ ∂ Γ = g(∇ , ) ij,k i ∂xj ∂xk (2) The Christoffel symbols can be computed as follows: 1 ∂g ∂g ∂g Γ = − ij + ik + jk , ij,k 2 ∂xk ∂xj ∂xi and k X km Γij = g Γij,m. m k k ∂ ∂ (3) We always have Γij,k = Γji,k and Γij = Γji since ∇i ∂xj = ∇j ∂xi in view of Axiom (6) of covariant derivative. P i ∂ P j ∂ (4) Let X = a ∂xi and Y = b ∂xj . Then 3 " # X X ∂bk X ∂ ∇ Y = ai + aibjΓk . X ∂xi ij ∂xk k i i,j Note that (∇X Y )|p depends only on X(p) and on the restriction of Y to some neighborhood of p on U (or even to some curve through p). 5. Covariant derivative for a vector field along a curve. Let c(t) be a smooth regular curve in M (here regular means that c˙(t) 6= 0 for every t). A vector field Y along c is a function Y (t) such that for every t we have Y (t) ∈ Mc(t). Let ∇ be a Riemannian connection on M corresponding to a Rie- mannian metric g. Then ∇c˙Y is a vector field along c defined as follows. Let (U, φ = (x1, . , xn)) be a chart. In this chart c(t) has the form (c1(t), . , cn(t)) where ci(t) = xi(c(t)), and Y has the form X ∂ Y (t) = bk(t) | . ∂xi c(t) k Then " # X X ∂ ∇ Y (t) = b˙k(t) + c˙i(t)bj(t)Γk (c(t)) | . c˙ ij ∂xk c(t) k i,j Note that this definition is chosen in such a way that if Y˜ is a vector ˜ field on M such that for every t we have Y (t) = Y |c(t) then for every t we have ˜ ∇c˙Y (t) = ∇c˙(t)Y. 6. Parallel vector fields and geodesics. Let (M n, g) be a smooth manifold with a smooth Riemannian metric g. Let ∇ be the unique Riemannian connection on M corresponding to g. Let c(t) be a smooth regular curve in M and let Y (t) be a vector field along c. We say that Y is parallel along c if ∇c˙Y = 0. We say that a smooth regular curve c is a geodesic in (M, g) ifc ˙ is parallel along c, that is, if ∇c˙c˙ = 0. (1) Let M 2 ⊆ R3 be an oriented surface in R3 as in Section 3 and let ∇ be a Riemannian connection on M given by the explicit formula from Section 3. Recall that Dc˙c˙ =c ¨. Then the condition ∇c˙c˙ = 0 is equivalent to 0 =c ¨ − hc,¨ nin, that is, thatc ¨ is parallel to n and thus perpendicular to Mc(t). 4 (2) If Y, Z are parallel along a regular curve c then g(Y (t),Z(t)) = const. In particular ||Y (t)|| = g(Y, Y )1/2 = const. Thus if c(t) is a geodesic, then ||c˙(t)|| = const. (3) Let (U, φ = (x1, . , xn) be a chart on M. Let X ∂ Y (t) = bk(t) | . ∂xi c(t) k and let ci(t) = xi(c(t)). Then Y is parallel along c if and only if ˙k X i j k b (t) + c˙ (t)b (t)Γij(c(t)) = 0, k = 1, . n. i,j (4) Let c(t) be a smooth regular curve as in (1) above. Then c is a geodesic if and only if it satisfies: k X i j k c¨ (t) + c˙ (t)˙c (t)Γij(c(t)) = 0, k = 1, . n. i,j 7. Facts about geodesics. Let (M, g) be a smooth manifold with a smooth Riemannian metric g and let p ∈ M. (1) For any v ∈ Mp, v 6= 0 there is some > 0 such that there exists a unique geodesic c :(−, ) → M with c(0) = p andc ˙(0) = v. (2) Let c1 : [0, a1] → M and c2 : [0, a2] → M be geodesics such that c1(0) = c2(0) = p andc ˙1(0) =c ˙2(0). Then c1(t) = c2(t) for every t ∈ [0, min{a1, a2}]. (3) If M is compact then for any v ∈ Mp, v 6= 0 there exists a unique geodesic c :(−∞, ∞) → M with c(0) = p andc ˙(0) = v. (4) There is some connected open set U in M containing p with the following properties: (a) For every q ∈ U, q 6= p there exists a unique unit speed geodesic cp,q connecting p to q. (b) For any other curve γ from p to q we have L(cp,q) ≤ L(γ). Moreover, if L(cp,q) = L(γ) and γ is a unit speed curve then γ = cp,q. (5) Let c : [0, a] → M be a regular smooth curve with c(0) = p. Then for any v ∈ Mp there exists a unique vector field Y (t) parallel along c such that Y (0) = v. In this case one says that w = Y (a) is obtained from v by a parallel transport along c..