Chapter 13 Curvature in Riemannian Manifolds 13.1 The Curvature Tensor If (M, , )isaRiemannianmanifoldand is a connection on M (that is, a connection on TM−), we− saw in Section 11.2 (Proposition 11.8)∇ that the curvature induced by is given by ∇ R(X, Y )= , ∇X ◦∇Y −∇Y ◦∇X −∇[X,Y ] for all X, Y X(M), with R(X, Y ) Γ( om(TM,TM)) = Hom (Γ(TM), Γ(TM)). ∈ ∈ H ∼ C∞(M) Since sections of the tangent bundle are vector fields (Γ(TM)=X(M)), R defines a map R: X(M) X(M) X(M) X(M), × × −→ and, as we observed just after stating Proposition 11.8, R(X, Y )Z is C∞(M)-linear in X, Y, Z and skew-symmetric in X and Y .ItfollowsthatR defines a (1, 3)-tensor, also denoted R, with R : T M T M T M T M. p p × p × p −→ p Experience shows that it is useful to consider the (0, 4)-tensor, also denoted R,givenby R (x, y, z, w)= R (x, y)z,w p p p as well as the expression R(x, y, y, x), which, for an orthonormal pair, of vectors (x, y), is known as the sectional curvature, K(x, y). This last expression brings up a dilemma regarding the choice for the sign of R. With our present choice, the sectional curvature, K(x, y), is given by K(x, y)=R(x, y, y, x)but many authors define K as K(x, y)=R(x, y, x, y). Since R(x, y)isskew-symmetricinx, y, the latter choice corresponds to using R(x, y)insteadofR(x, y), that is, to define R(X, Y ) by − R(X, Y )= + . ∇[X,Y ] ∇Y ◦∇X −∇X ◦∇Y As pointed out by Milnor [106] (Chapter II, Section 9), the latter choice for the sign of R has the advantage that, in coordinates, the quantity, R(∂/∂x ,∂/∂x)∂/∂x ,∂/∂x coincides h i j k 399 400 CHAPTER 13. CURVATURE IN RIEMANNIAN MANIFOLDS with the classical Ricci notation, Rhijk. Gallot, Hulin and Lafontaine [60] (Chapter 3, Section A.1) give other reasons supporting this choice of sign. Clearly, the choice for the sign of R is mostly a matter of taste and we apologize to those readers who prefer the first choice but we will adopt the second choice advocated by Milnor and others. Therefore, we make the following formal definition: Definition 13.1 Let (M, , )beaRiemannianmanifoldequippedwiththeLevi-Civita connection. The curvature− tensor− is the (1, 3)-tensor, R,definedby R (x, y)z = Z + Z Z, p ∇[X,Y ] ∇Y ∇X −∇X ∇Y for every p M and for any vector fields, X, Y, Z X(M), such that x = X(p), y = Y (p) and z = Z(p∈). The (0, 4)-tensor associated with R,alsodenoted∈ R,isgivenby R (x, y, z, w)= (R (x, y)z,w , p p for all p M and all x, y, z, w T M. ∈ ∈ p Locally in a chart, we write ∂ ∂ ∂ l ∂ R , = Rjhi ∂xh ∂xi ∂xj ∂xl l and ∂ ∂ ∂ ∂ l Rhijk = R , , = glkRjhi. ∂xh ∂xi ∂xj ∂xk l l k The coefficients, Rjhi,canbeexpressedintermsoftheChristoffelsymbols,Γij,intermsofa rather unfriendly formula (see Gallot, Hulin and Lafontaine [60] (Chapter 3, Section 3.A.3) or O’Neill [119] (Chapter III, Lemma 38). Since we have adopted O’Neill’s conventions for l the order of the subscripts in Rjhi, here is the formula from O’Neill: Rl = ∂ Γl ∂ Γl + Γl Γm Γl Γm. jhi i hj − h ij im hj − hm ij m m There is another way of defining the curvature tensor which is useful for comparing second covariant derivatives of one-forms. Recall that for any fixed vector field, Z,themap, Y Y Z,isa(1, 1) tensor that we will denote Z.Thus,usingProposition11.5,the → ∇ ∇− covariant derivative X Z of Z makes sense and is given by ∇ ∇− ∇− ( X ( Z))(Y )= X ( Y Z) ( Y )Z. ∇ ∇− ∇ ∇ − ∇∇X 2 Usually, ( X ( Z))(Y )isdenotedby X,Y Z and ∇ ∇− ∇ 2 X,Y Z = X ( Y Z) X Y Z ∇ ∇ ∇ −∇∇ 13.1. THE CURVATURE TENSOR 401 is called the second covariant derivative of Z with respect to X and Y .Then,wehave 2 2 Y,XZ X,Y Z = Y ( X Z) Y X Z X ( Y Z)+ X Y Z ∇ −∇ ∇ ∇ −∇∇ −∇ ∇ ∇∇ = Y ( X Z) X ( Y Z)+ Y X Z ∇ ∇ −∇ ∇ ∇∇X −∇Y = ( Z) ( Z)+ Z ∇Y ∇X −∇X ∇Y ∇[X,Y ] = R(X, Y )Z, since X Y Y X =[X, Y ], as the Levi-Civita connection is torsion-free. Therefore, the curvature∇ tensor−∇ can also be defined by R(X, Y )Z = 2 Z 2 Z. ∇Y,X −∇X,Y We already know that the curvature tensor has some symmetry properties, for example, R(y, x)z = R(x, y)z but when it is induced by the Levi-Civita connection, it has more remarkable properties− stated in the next proposition. Proposition 13.1 For a Riemannian manifold, (M, , ), equipped with the Levi-Civita connection, the curvature tensor satisfies the following− properties:− (1) R(x, y)z = R(y, x)z − (2) (First Bianchi Identity) R(x, y)z + R(y, z)x + R(z,x)y =0 (3) R(x, y, z, w)= R(x, y, w, z) − (4) R(x, y, z, w)=R(z,w,x,y). The proof of Proposition 13.1 uses the fact that Rp(x, y)z = R(X, Y )Z,foranyvector fields X, Y, Z such that x = X(p), y = Y (p)andZ = Z(p). In particular, X, Y, Z can be chosen so that their pairwise Lie brackets are zero (choose a coordinate system and give X, Y, Z constant components). Part (1) is already known. Part (2) follows from the fact that the Levi-Civita connection is torsion-free. Parts (3) and (4) are a little more tricky. Complete proofs can be found in Milnor [106] (Chapter II, Section 9), O’Neill [119] (Chapter III) and Kuhnel [91] (Chapter 6, Lemma 6.3). If ω 1(M)isaone-form,thenthecovariantderivativeofω defines a (0, 2)-tensor, T , ∈A 2 given by T (Y,Z)=( Y ω)(Z). Thus, we can define the second covariant derivative, X,Y ω, of ω as the covariant∇ derivative of T (see Proposition 11.5), that is, ∇ ( T )(Y,Z)=X(T (Y,Z)) T ( Y,Z) T (Y, Z), ∇X − ∇X − ∇X and so 2 ( X,Y ω)(Z)=X(( Y ω)(Z)) ( X Y ω)(Z) ( Y ω)( X Z) ∇ ∇ − ∇∇ − ∇ ∇ = X(( Y ω)(Z)) ( Y ω)( X Z) ( Y ω)(Z) ∇ − ∇ ∇ − ∇∇X =(X ( Y ω))(Z) ( Y ω)(Z). ∇ ∇ − ∇∇X 402 CHAPTER 13. CURVATURE IN RIEMANNIAN MANIFOLDS Therefore, 2 X,Y ω = X ( Y ω) X Y ω, ∇ ∇ ∇ −∇∇ that is, 2 ω is formally the same as 2 Z.Then,itisnaturaltoaskwhatis ∇X,Y ∇X,Y 2 ω 2 ω. ∇X,Y −∇Y,X The answer is given by the following proposition which plays a crucial role in the proof of a version of Bochner’s formula: Proposition 13.2 For any vector fields, X, Y, Z X(M), and any one-form, ω 1(M), on a Riemannian manifold, M, we have ∈ ∈A (( 2 2 )ω)(Z)=ω(R(X, Y )Z). ∇X,Y −∇Y,X Proof . Recall that we proved in Section 11.5 that ( ω) = ω. ∇X ∇ We claim that we also have ( 2 ω) = 2 ω. ∇X,Y ∇X,Y This is because 2 ( X,Y ω) =(X ( Y ω)) ( X Y ω) ∇ ∇ ∇ − ∇∇ = X ( Y ω) Y ω ∇X ∇ ∇ −∇ = X ( Y ω ) Y ω ∇ ∇ −∇∇X = 2 ω. ∇X,Y Thus, we deduce that (( 2 2 )ω) =( 2 2 )ω = R(Y,X)ω. ∇X,Y −∇Y,X ∇X,Y −∇Y,X Consequently, (( 2 2 )ω)(Z)= (( 2 2 )ω),Z ∇X,Y −∇Y,X ∇X,Y −∇Y,X = R(Y,X)ω,Z = R(Y,X,ω,Z) = R(X, Y, Z, ω) = R(X, Y )Z,ω = ω(R(X, Y )Z), where we used properties (3) and (4) of Proposition 13.1. The next proposition will be needed in the proof of the second variation formula. If α: U M is a parametrized surface, where U is some open subset of R2,wesaythata → 13.2. SECTIONAL CURVATURE 403 vector field, V X(M), is a vector field along α iff V (x, y) Tα(x,y)M,forall(x, y) U. For any smooth∈ vector field, V ,alongα,wealsodefinethecovariantderivatives,∈ DV/∈ ∂x and DV/∂y as follows: For each fixed y0,ifwerestrictV to the curve x α(x, y ) → 0 we obtain a vector field, Vy0 ,alongthiscurveandweset DX DV (x, y )= y0 . ∂x 0 dx Then, we let y vary so that (x, y ) U and this yields DV/∂x.WedefineDV/∂y is a 0 0 ∈ similar manner, using a fixed x0. Proposition 13.3 For a Riemannian manifold, (M, , ), equipped with the Levi-Civita − − connection, for every parametrized surface, α: R2 M, for every vector field, V X(M) along α, we have → ∈ D D D D ∂α ∂α V V = R , V. ∂y ∂x − ∂x ∂y ∂x ∂y Proof . Express both sides in local coordinates in a chart and make use of the identity ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = R , . ∇ ∂xj ∇ ∂xi ∂x −∇∂xi ∇ ∂xj ∂x ∂x ∂x ∂x k k i j k Remark: Since the Levi-Civita connection is torsion-free, it is easy to check that D ∂α D ∂α = . ∂x ∂y ∂y ∂x We used this identity in the proof of Theorem 12.18. The curvature tensor is a rather complicated object. Thus, it is quite natural to seek simpler notions of curvature.