The Riemannian Connection • If We Are to Use Geodesics and Covariant

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.

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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 tangential 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 symmetric. 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 ì ` 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 ì ` 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)|. • We say γ is of constant speed if |γ0(t)| is independent of t. • We say γ is of unit speed if |γ0(t)| =1for all t. Lemma 6. All Riemannian geodesics are constant speed curves. Proof. Let γ be a Riemannian geodesic. Since γ0 is parallel along γ, its length γ0 = hγ0,γ0i1/2 is constant by Lemma 3(d).

Proposition 7 (Naturality of the Riemannian Connections). Suppose ϕ : (M,g) → (M,g) is an isometry. f (a) ϕ takes the Riemannian connection D of g to the Riemannian connection D of e g, in the sense that e ϕ∗(D Y )=D (ϕ∗Y ). X eϕ∗X (b) If γ is a curve in M and V is a vector ﬁeld along γ, then

ϕ∗D V = D (ϕ∗V ). t et (c) ϕ takes geodesics to geodesics: if γ is the geodesic in M with initial point p and initial velocity V , then ϕ ◦ γ is the geodesic in M with initial point ϕ(p) and f initial velocity ϕ∗V . Proof. (a) Deﬁne a map ϕ∗D :Γ(TM) × Γ(TM) → Γ(TM) e by ∗ −1 (ϕ D) Y = ϕ (D (ϕ∗Y )). e X ∗ eϕ∗X Then ϕ∗∇˜ is a connection on M (called the pullback connection), and it is symmetric and compatible; therefore ϕ∗D = D e by uniqueness of the Riemannian connection. (b) Deﬁne an operator ϕ∗D : T (γ) →T(γ) by a similar formula which is equal to et Dt .

• The reason why this connection has been anited as “the” Riemannian connection is: symmetry and compatibility are invariantly deﬁned and natiral properties that force the connection to coincide with the tangential connection whenever M is realized as a submanifold of Rn with the induced metric (which the Nash embedding theorem guarantees is always possible).