Let M be a Riemannian manifold with Levi-Civita connection D. For X, Y, W Γ(TM), we have R(X, Y )Z = D Y D X Z D X D Y Z D [X,Y ] Z,

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1 Let M be a Riemannian manifold with Levi-Civita connection D. For X, Y, W Γ(TM), we have R(X, Y )Z = D Y D X Z D X D Y Z D [X,Y ] Z, where R is the curvature tensor of D. We have Proposition 1. For any vector fields W, X, Y, Z: (a) R(X, Y )Z = R(Y,X)Z. (b) R(X, Y )Z + R(Z, X)Y + R(Y,Z)X =0. (c) R(X, Y )Z, W + R(X, Y )W, Z =0. (d) R(X, Y )Z, W R(Z, W )X, Y =0. Proposition 2. For p M, ξ, η T p M, define k(ξ,η) = R(ξ,η)ξ,η. Then, we have for any ξ, η, ν T p M R(ξ,η)ξ,η = s t {k(ξ + sη, η + tµ) k(ξ + sµ, η + tζ)}. s=t=0 Thus, the function k : T p M T p M R and the properties (a), (b), (c) and (d) in Proposition 1 completely determine R : T p M T p M T p M T p M. Proof. Direct calculation using (a), (b), (c) and (d). We have {k(ξ + sη, η + tµ) k(ξ + sν, η + tζ)} = R(ξ + sζ, η + tµ)ξ + sζ, η + tµ R(ξ + sµ, η + tζ)ξ + sµ, η + tζ =As 2 + Bst + Ct 2, and we have 2 s t {k(ξ + sη, η + tµ) k(ξ + sµ, η + tζ)} = B. s=t=0 We proceed to compute B: B = R(ξ,η)ζ,µ + R(ξ,µ)ζ,η + R(ζ,µ)ξ,η + R(ζ,η)ξ,µ, R(ξ,η)µ, ζ R(ξ,ζ)µ, η R(µ, η)ξ,ζ R(µ, ζ)ξ,η, =2 R(ξ,η)ζ,µ +2 R(ζ,µ)ξ,η + R(ξ,µ)ζ,η + R(ζ,η)ξ,µ R(ξ,ζ)µ, η R(µ, η)ξ,ζ =4 R(ξ,η)µ, ζ + 2 R(ξ,µ)ζ, η + 2 R(ζ, ξ)µ, η, by (d), =4 R(ξ,η)µ, ζ + 2 R(µ, ζ)ξ,η, by (b) =6 R(ξ,η)µ, ζ. Typeset by AMS-TEX 1

2 2 Proposition 3. For p M, ξ, η, ζ T p M, define R 1 (ξ,η)ζ = ξ,ζ η η, ζ ξ, k 1 (ξ,η) = R 1 (ξ,η)ξ,η. Then k 1 (ξ,η) = ξ,ξ η, η ξ,η 2, and R 1 satisfies the axioms (a), (b), (c), (d). Furthermore, if ξ, η are linearly independent tangent vectors in T p M, then (1) K(ξ,η) =: k(ξ,η) k 1 (ξ,η) = R(ξ,η)ξ,η ξ 2 η 2 ξ,η 2 is well-defined and only depends on the 2-dimensional subspace determined by ξ and η. Proof. To prove the last claim, it suffices to note that if α, β, γ, δ R and ξ, η linearly, independent tangent vectors in T p M, then and R(αξ + βη,γξ + δη)αξ + βη,γξ + δη =(αδ βγ) 2 R(ξ,η)ξ,η, αξ + βη 2 γξ + δη 2 αξ + βη,γξ + δη 2 =(αδ βγ) 2 { ξ 2 η 2 ξ,η }. Definition. K(ξ,η) is called the sectional curvature of the 2-section determined by ξ, η. Definition. G 2 is the collection of all 2-dimensional spaces tangent to M. (G 2 can be provided a smooth structure, and K : G 2 R will then be smooth.) If dim M =2, then G 2 = M; K is called the Gauss curvature of M. Theorem 4. If dim M =2, for p M, ξ, η, ζ T p M, we have R(ξ,η)ζ = K(p)R 1 (ξ,η)ζ. Proof. Let {e 1,e 2 } be an orthonormal basis of T p M. We only have to verify (1) for ξ = ζ = e 1, η = e 2. To this end, consider the map R : T p M T p M, defined by R(ξ) =R(e 1,ξ)e 1. Then, by (c) in Proposition 1, R is self-adjoint and therefore diagonalizable. Since e 1 is an eigenvector of R with eigenvector 0, we have that e 2 is an eigenvalue of R with eigenvalue Thus, Re 2,e 2 = R(e 1,e 2 )e 1,e 2 = K(p). R(e 1,e 2 )e 1 = K(p)e 2 = K(p)R 1 (e 1,e 2 )e 1.

3 Theorem 5. A Riemannian manifold M is of constant sectional curvature C iff the following identity holds: (2) R(ξ,η)ζ = C{ η, ζ ξ ζ,ξ η}, ξ,η,ζ T p M, p M. 3 Proof. ( ) Obvious. ( ) Suppose M is of constant sectional curvature C. Setting S(ξ,η)ζ = R(ξ,η)ζ C{ η, ζ ξ ζ,ξ η}, we see that S is a tensor of type ( 3 1) which satisfies the algebraic conditions (a), (b), (c) and (d) of the curvature tensor. (i) Now from the assumption we claim: (3) S(ξ,η)η, ξ =0, ξ,η T p M, p M. - Indeed, this is obvious if ξ and η are linearly dependent. - If ξ and η are linearly independent, then (3) follws from (1). (ii) Now insertng η + ζ instead of η in (3) and using (3), (a), (c), (d), we obtain that is, S(ξ,η)ξ,ζ =0, ξ,η,ζ T p M, p M, S(ξ,η)ξ =0, ξ,η T p M. (iii) Again inserting ξ + µ instead of ξ in the last equation, we obtain S(ξ,η)µ + S(µ, η)ξ =0. On the other hand, by the first Bianchi identity (1), S(ξ,η)µ = S(η, µ)ξ S(µ, ξ)η. These two identities imply 2S(µ, η)ξ = S(µ, ξ)η. Replacing ξ, η we obtain S(µ, ξ)η =0.

4 4 Ricci and Scalar Curvatures Beacuse 4-tensors are so complicated, it is often useful to construct simpler tensors that summarize some of the information obtained in the curvature tensor. Definition. The Ricci curvature or Ricci tensor, denoted Rc, is the covariant 2-tensor field Ric : Γ(TM) Γ(TM) R defined as the trace of the curvature endomorphism. Ric(ξ,η) =trace(ζ R(ξ,ζ)η). The components of Ric are usually denoted R ij, so that R ij = R ikj k = g km R ikjm. Lemma. The Ricci curvature is a symmetric 2-tensor field. It can be expressed in any of the following ways: R ij = R k ikj = Rki k j = R kij k. In particular, we have for any orthonormal basis {e 1,,e n } of T p M, Ric(ξ,η) = n R(ξ,e j )η, e j = Ric(η, ξ), j=1 Thus, Ric is a symmetric bilinear form on T p M. To calculate its associated bilinear form, choose {e 1,,e n } so that e n = ξ/ ξ, and then { n 1 } Ric(ξ,ξ) = K(e j,ξ) ξ 2. j=1 Definition. The scalar curvature is the function S defined as the trace of the Ricci tensor S = tr g Ric = R i i = g ij R ij. For any orthonormal basis {e 1,,e n } of T p M, we have n S = K(e j,e k ). j =k;j,k=1

5 Definition. A Riemannian metric is said to be an Einstein metric if its Ricci tensor is a scalar multiple of the metric at each point; that is, for some function λ, Ric = λg everywhere. Taking traces of both sides and noting that tr g g = g ij g ij = δ i i = dimm, 5 we find that λ = 1 S (where n =dim M). n Thus the Einstein condition can be written (4) Ric = 1 n Sg. Hilbert showed that Einstein metrics are critical points for the total curvature functional S(g) = M SdV on the space of all metrics on M with fixed volume. Einstein metrics can be viewed as optimal metrics in a certain sense, and as such they form an appealing higher-dimensional analogue of the metrics of constant Gaussian curvature on 2-manifolds, with which one might hope to prove some sort of generalization of the uniformization theorem. Although the statement of such a theorem cannot be as elegant as that of its 2-dimensional ancestor because there are known examples of smooth, compact manifolds that admit no Einstein metrics, there is still a reasonable hope that most higher-dimensional manifolds (in some sense) admit Einstein metrics. Another approach to generalizing the uniformization theorem to higher dimensions is to search for metrics of constant scalar curvature. These are also critical points of the total scalar curvature functional, but only w.r.t. variations of the metric within a given equivalence class. Thus it makes sense to ask whether, given a metric g on a manifold M, there exists a metric g conformal to g that has constant scalar curvature. This is called the Yamabe problem.