The Riemannian Connection • If we are to use geodesics and covariant derivatives as tools for studying Rie- mannian geometry, it is evident that we need a way to single out a particular connection on a Riemannian manifold that reflects the properties of the metric. • We are going to show that on each Riemannian manifold there is a natural connection that is particularly suited to computations in Riemannian geometry. • Guided by an example of an embedded submfd of Rn, we describe two properties that determine a unique connection on any Riemannian manifold. (1) The first property, compatibility with the metric, is easy to motivate and understand. (2) The second, symmetry, is a bit more mysterious. • Since we get most of our intuition about Riemannian manifolds from studying submanifolds of Rn with the induced metric, let us start by examining that case. • As a guiding principle, consider the idea: A geodesic in a submfd of Rn should be “as straight as possible”, which we take to mean that its acceleration vector field should have zero tangential projection onto TM. • To express this in the language of connections, let M ⊂ Rn be an embedded submanifold. Any vector field on M can be extended to a smooth vector field on Rn. (Why?) • Begin with a trivial example on Rn. Definition. Define the Euclidean connection by j j DX (Y ∂j )=(XY )∂j . — In other words, DX Y is just the vector field whose components are the ordinary directional derivatives of the components of Y in the direction X. — It is easy to check that this satisfies the required properties for a connection, and that its Christoffel symbols in standard coordinates are all zero. • Define a map D> :Γ(TM) × Γ(TM) → Γ(TM) by setting > > DX Y = π (DX Y ), where X and Y are extended arbitrarily to Rn, D is the Euclidean connection n n on R , and for any point p ∈ M, π : TpR → TpM is the orthogonal projection. • As the next lemma shows, this turns out to be a connection on M, called the tangential connection. Typeset by AMS-TEX 1 2 Lemma 1. The operator D> is well-defined, and is a connection on M. > Proof. (1) Claim: DX Y is well-defined. > (i) DX Y is clearly independent of the choice of vector fields extending X, since the value of DX Y at a point p ∈ M depends only on X. (ii) The value of DX Y at p depends only on the values of Y along a curve whose initial tangent vector is Xp; taking the curve to lie entirely in M shows that > DX Y depends only on the original vector field Y ∈ Γ(TM). (2) Smoothness follows easily by expressing DX Y in terms of an adapted orthonor- mal frame. > ∞ (3) It is obvious from the definition that DX Y is linear over C (M)inX and over R in Y . (4) To check the product rule, let f ∈ C∞(M) be extended arbitrarily to Rn. Evaluating along M, we obtain > > DX (fY)=π (DX (fY)) > > =(Xf)π Y + fπ (DX Y ) > =(Xf)Y + fDX Y. • A theorem of John Nash says that any Riemannian metric on any manifold can be realized as the induced metric of some embedding in a Euclidean space. — Thus, in a certain sense, one would lose no generality by studying only submfds of Rn with their induced metrics, for which the tangential connection would suffice. • However, when one is trying to understand intrinsic properties of a Riemannian manifolds, an embedding introduces a great deal of extraneous information, and in some cases actually makes it harder to discern which geometric properties depend only on the metric. • Our task in this chapter is to distinguish some important properties of the tan- gential connection that make sense for connections on an abstract Riemannian manifold, and to use them to single out a unique connection in the abstract case. • The Euclidean connection on Rn has one very nice property w.r.t. the Euclidean metric: it satisfies the product rule DZ hX, Y i = hDX Y,Zi + hY,DX Zi, as we can verify easily by computing in terms of the standard basis. — It is now immediate that the tangential connection has the same property,ifwe now interpret all the vector fields as being tangent to M and interpret the inner product as being taken w.r.t. the induced metric on M. Definition. Let g be a Riemannian (or pseudo-Riemannian) metric on a manifold M. A connection D is said to be compatible with g if it satisfies the following product rule for all vector fields X, Y , and Z. DZ hX, Y i = hDX Y,Zi + hY,DX Zi. 3 Lemma 3. The following conditions are equivalent for a onnection D on a Rie- mannian manifold: (1) D is compatible with g. (2) Dg ≡ 0. (3) If V , W are vector fields along any curve γ, d hV,Wi = hD V,Wi + hV,D W i. dt t t (4) If V , W are parallel vector fields along a curve γ, then hV,Wi is constant. → ∀ (5) Parallel translation Pt0t1 : Tγ(t0)M Tγ(t1)M is an isometry t0, t1. • It turns out that requring a connection to be compatible with the metric is not enough to determine a unique connection, so we turn to another key property of the tangential connection. This involves the torsion tensor of the connection. 2 Definition. The torsion tensor of the connection is the 1-tensor field τ : Γ(TM) × Γ(TM) → Γ(TM) defined by τ(X, Y )=DX Y − DY X − [X, Y ]. • A linear connection ∇ is said to be torsion-free or symmetric if its torsion vanishes identically; i.e., if DX Y − DY X ≡ [X, Y ]. Lemma 4. The tangential connection on an embedded submfd M ⊂ Rn is sym- metric. Theorem 5 (Fundamental Lemma of Riemannian Geometry). Let (M,g) be a Riemannian (or pseudo-Riemannian) manifold. There exists a unique linear connection D on M that is compatible with g and symmetric. • This connection is called the Riemannian connection or the Levi-Civita connection of g. Proof. We prove uniqueness first, by deriving a formula for D. Let D be such a connection, and let X, Y , Z ∈ Γ(TM) be arbitrary vector fields. Write the compatibility equation three times with X, Y , Z cyclically permuted: XhY,Zi = hDX Y,Zi + hY,DX Zi, Y hZ, Xi = hDY Z, Xi + hZ, DY Xi, ZhX, Y i = hDZ X, Y i + hX, DZ Y i. Using the symmetry condition on the last term in each line, this is rewritten as 4 XhY,Zi = hDX Y,Zi + hY,DZ Xi + hY,[X, Z]i, Y hZ, Xi = hDY Z, Xi + hZ, DX Y i + hZ, [Y,X]i, ZhX, Y i = hDZ X, Y i + hX, DY Zi + hX, [Z, Y ]i. — Adding the first two of these equations and subtracting the third, we obtain XhY,Zi+Y hZ, Xi−ZhX, Y i =2hDX Y,Zi+hY,∇Z Xi+hZ, [Y,X]i−hX, [Z, Y ]i. Hence 1 (1) hD Y,Zi = (XhY,Zi+Y hZ, Xi−ZhX, Y i) X 2 −hY,[X, Z]i−hZ, [Y,X]i + hX, [Z, Y ]i). — Now suppose D1 and D2 are two connections that are symmetric and compatible with g, Since the right-hand side of (1) does not depend on the connection, it follows that 1 2 hDX Y − DX Y,Zi =0, ∀X,Y,Z. 1 2 1 2 This can only happen if DX Y = DX Y for all X and Y , and so D = D . • Existence: We use (1) or rather a coordinate version of it. — It suffices to prove that such a connection exists in each coordinate chart, for then uniqueness ensures that the connection constructed in different charts agree where they overlap. — Let (U, (xi)) be any local coordinate chart. Applying (1) to the coordinate vector fields, whose Lie brackets are zero, we obtain 1 (2) hD ∂ ,∂ i = (∂ h∂ ,∂ i + ∂ h∂ ,∂ i−∂ h∂ ,∂ i. ∂i j ` 2 i j ` j ` i ` i j Recall the definition of the metric coefficients and the Christoffel symbols: m gij = h∂i,∂j i,D∂i ∂j =Γij ∂m. Inserting these into (2) yields 1 (3) Γmg = (∂ g + ∂ g − ∂ g ). ij m` 2 i j` j `i ` ij ik `k k Multiplying both sides by the inverse matrix g and noting that gm`g = δm, we obtain 1 (4) Γk = gk`(∂ g + ∂ g − ∂ g ). ij 2 i j` j `i ` ij — This formula certainly defines a connection in each chart. k k — It is evident from the formula that Γij =Γij , so the connection is symmetric. — Thus only compatibility with the metric needs to be checked. By Lemma 3, it suffices to claim: Dg =0. – In terms of a coordinate frame, the components of Dg are ` ` gij;k = ∂kgij − Γkig`j − Γkjgi`, Using (3) twice, we conclude 1 1 Γ` g +Γ` g = (∂ g + ∂ g − ∂ g )+ (∂ g + ∂ g − ∂ g )=∂ g , ki ij kj i` 2 k ij i kj j ki 2 k ji j ki i kj k ij which shows that gij;k =0. 5 • A bonus of this proof is that it gives us an explicit formula (4) for computing the Christoffel symbols of the Riemannian connection in any coordinate chart. • On any Riemannian mfd, we will always use the Riemannian connection from now on without further comment. Definition. Geodesics with respect to the Riemannian connection are called Rie- mannian geodesics, or simply geodesics.. Definition. If γ is a curve in a Riemannian mfd, the speed of γ at any time t is the length of its velocity vector |γ0(t)|.