Riemannian Manifolds

Transcription

1 Chapter 25 Riemannian Manifolds Our ultimate goal is to study abstract surfaces that is 2-dimensional manifolds which have a notion of metric compatible with their manifold structure see Definition 2521 on page 858 The theory of differentiable manifolds developed in the previous chapter is insufficient on its own to measure lengths and areas For that we need to impose additional structure so that we end up with a generalization of a metric as defined in Section 121 Riemann 1 in his thesis [Riem] written in 1854 but published only in 1867 introduced such a metric ds 2 in n dimensions and described how the curvature of ds 2 can be measured In this chapter we develop key concepts like that of a covariant derivative and a pseudo-riemannian metric that bolt on to the definition of differentiable manifold It is just as easy to start by defining these concepts for arbitrary dimensions which is what we do first though we shall concentrate on 2 dimensions in the last two chapters of this book Covariant derivatives are defined and their basic properties are developed in Section 251 They will play a particularly important role in the further discussion of geodesics in Chapter 26 The theory of covariant derivatives or connec- 1 Georg Friedrich Bernhard Riemann German mathematician Although he died young and published few papers Riemann s ideas have had a profound effect on the development of mathematics and physics In an early paper he clarified the notion of integral by defining what is now called the Riemann integral Riemann s Habilitationsvortrag [Riem] Über die Hypothesen welche der Geometrie zu Grunde liegen impressing even Gauss became a classic and its results were incorporated into Einstein s general relativity The Riemann hypothesis which concerns the zeros of the complex analytic function ζz n1 1 n z has enormous implications for number theory but remains unsolved 847

2 848 CHAPTER 25 RIEMANNIAN MANIFOLDS tions as they are called in a more abstract setting is one of the cornerstones of contemporary research in differential geometry The notion was introduced by Levi-Civita 2 in his book The Absolute Differential Calculus [LeviC] to explain parallelism in Riemannian manifolds Pseudo-Riemannian manifolds are defined in Section 252; it is a significant fact that every pseudo-riemannian manifold has a natural covariant derivative Definition 2518 A Riemannian manifold is a pseudo-riemannian manifold whose metric is positive definite Section 253 is devoted to bridging the gap between the more modern techniques of Sections 251 and 252 on the one hand and the classical treatment of metrics on the other We begin the final section by making precise the way in which the Christoffel symbols fail to be the components of a tensor We then define the celebrated Riemann curvature tensor R and state a number of standard results deferring proofs to the Exercises When R is expressed in terms of coordinates one recognizes formulas from the discussion of the Peterson Mainardi Codazzi equations in Section 193 This will enable us to pin down the precise relationship between the Riemann and Gaussian curvature for an abstract surface in the next chapter 251 Covariant Derivatives Since it is important to study vector fields on a differentiable manifold M it is natural to try to differentiate vector fields on M For R n there is an obvious way to do this Recall that XR n denotes the space of differentiable vector fields defined on R n Examples are the vector fields 251 u 1 u n determined by the natural coordinate functions u 1 u n Now consider an arbitrary vector field Y XR n ; in terms of 251 we can write Y f i u i for some functions f 1 f n FR n As a matter of fact in calculus the vector field Y is frequently identified with the n-tuple f 1 f n The obvious candidate for the derivative of Y in the direction of a vector X is then 2 Tullio Levi-Civita Italian mathematician professor at Padua and Rome He also wrote on hydrodynamics and rational mechanics Levi- Civita was an outspoken opponent of fascism and was forced to give up teaching in 1938

3 251 COVARIANT DERIVATIVES D X Y X[f 1 ] X[f n ] as defined in Lemma 932 on page 277 In the more abstract setting we are about to consider the same quantity will be denoted by the slightly different notation D X Y as a reminder of the fact that differentiation D no longer takes place in a Euclidean space For a general differentiable manifold there is no automatic way to differentiate vector fields Instead we must add a new element of structure: Definition 251 A connection or covariant derivative on a differentiable manifold M is a map D: XM XM XM which for X XM also defines a map D X : XM XM by setting DXY D X Y It is required to have the properties: D fx+gy fd X + gd Y D X Y + Z D X Y + D X Z D X fy X[f]Y + fd X Y for XYZ XM and f g FM Remarks 1 A connection D is not a tensor field in the sense of Section 246 Despite the linearity condition 253 it is not linear with respect to functions in its second argument 2 A connection is however linear with respect to real numbers in its second argument as equations 254 and 255 imply that 256 D X ay + bz ad X Y + bd X Z for a b R and XYZ XM 3 Using a partition of unity it is possible to show that every differentiable manifold has a connection though there is in general no canonical choice We shall see that every Riemannian metric has a special connection associated with it

4 850 CHAPTER 25 RIEMANNIAN MANIFOLDS Example: The connection on Euclidean space Define D: XR n XR n XR n by first putting D u i for 1 i j n More generally for Y 0 u j f i u i we are now forced by 255 and 256 to require that 257 D X Y X[f i ] u i for 1 i n In fact 257 defines a natural connection on R n ; moreover 257 is equivalent to 252 There is a variant of a connection analogous to the definition of D v that we gave in Section 95 Let M be a differentiable manifold and let p M A connection D on M gives rise to a map defined as follows D: M p XM M p Definition 252 Let v be a tangent vector to M at p M Choose a vector field V XM such that V p v For W XM we define D v W to be the tangent vector in M p given by D v W D V W p Lemma 253 Fix W XM The definition of D v W is independent of the choice of V XM such that V p v Proof Let VY XM be such that V p Y p and let x 1 x n be a coordinate system defined in an open set containing p By Corollary 2455 page 834 we can write V V[x i ] and Y x i Y[x i ] x i

5 251 COVARIANT DERIVATIVES 851 Then 253 and 255 imply that D V W p D Y W p V[x i ]D /xi W p V[x i ]p D /xi W p Y[x i ]p D /xi W p It is important to realize that the statement of the previous lemma refers to the first argument of D v w In general it is not true that Y p Z p implies D X Y p D X Z p We can however make a useful statement using the next concept Definition 254 Let X XM An integral curve of the vector field X is any curve α: a b M for which α t X αt for all t a b The statement which we are after deals with the case in which two vector fields agree along an integral curve: Lemma 255 Let X XM and let α: a b M be an integral curve of X If YZ XM satisfy Y αt Z αt whenever a < t < b then for a < t < b D X Y αt D X Z αt Proof We can assume without loss of generality that there is a system of coordinates x 1 x n defined on an open set W M such that the trace of α is contained in W It suffices to show that for any Y XM if Y αt vanishes identically then so does D X Y αt By Corollary 2455 we can write 258 Y Y[x i ] x i on W Suppose that t Y αt vanishes identically Then 258 implies that Y[x i ] αt 0 for a < t < b and i 1 n When we differentiate 258 we obtain D X Y XY[x i ] + Y[x i ]D x X i x i

6 852 CHAPTER 25 RIEMANNIAN MANIFOLDS which restricts to α as 259 D X Y αt But by Lemma 2439 we have XY[x i ] x i αt XY[x i ] αt X αt [ Y[xi ] ] α t [ Y[x i ] ] dy[x i] α dt 0 Thus the coefficients in 259 all vanish and so D X Y αt 0 for a < t < b In Section 12 we considered vector fields along a curve in R n This notion generalizes to manifolds Definition 256 Let α: a b M be a curve in a differentiable manifold M A vector field along α is a function Y that assigns to each t a b a tangent vector Yt M αt Lemma 255 allows us to define unambiguously the covariant derivative of a vector field along a curve: Definition 257 Let α: a b M be an injective regular curve in a differentiable manifold M and let Y be a vector field along α Then Y t is the vector field along α defined by Y t D α tỹ where Ỹ XM is any vector field such that Ỹαt Yt for a < t < b Vector fields along a curve in a manifold M have most of the properties enjoyed by vector fields along a curve in R n For example the following lemma generalizes Lemma 111 on page 8 Lemma 258 Let Y and Z be vector fields along a curve α: a b M in a differentiable manifold M and let f : a b R be differentiable Then Y + Z Y + Z 2510 fy f Y + fy Proof We prove the second equation 2510 Let f FM and Ỹ XM be such that fαt ft and Ỹαt Yt for a < t < b Then fy t D α f Ỹ t α t[ f]ỹαt + f αt D α tỹ f α tyt + fty t f tyt + fty t There is also a chain rule for vector fields along a curve

7 251 COVARIANT DERIVATIVES 853 Lemma 259 Suppose that Y is a vector field along a curve α: a b M in a differentiable manifold M and let h: c d a b be differentiable Then 2511 for c < t < d Y h t h ty ht Proof If Ỹ XM is such that Ỹ αht Yht for c < t < d then by the chain rule for curves Lemma 2441 on page 831 we have Y h t D α h tỹ D h tα htỹ h td α htỹ h ty ht Now we are ready to define the notion of acceleration of a curve in a manifold It depends on the choice of covariant derivative Definition 2510 Let α: a b M be a curve in a differentiable manifold M and let D be a connection on M The acceleration of α with respect to D is the vector field t D α t α t which we write more simply as D α α We say that a curve α: a b M is a geodesic with respect to D provided for a < t < b D α t α t 0 We have denoted the acceleration of a curve α in a manifold M by D α α to avoid confusion with the acceleration of a curve in R n For example if α is a curve in a surface M R n the two accelerations D α α and α are usually different This is explained in Section 263 in the next chapter which includes a justification for the general definition of geodesic which we have just given Next we prove the manifold version of equation 116 on page 16 which tells us the relation between the acceleration of a curve α and the acceleration of a reparametrization of α Lemma 2511 Let D be a connection on a differentiable manifold M and let α: a b M be a curve on M Suppose that β α h is a reparametrization of α where h: c d a b is differentiable Then 2512 D β β D α α h h 2 + α hh

8 854 CHAPTER 25 RIEMANNIAN MANIFOLDS Proof From Lemma 2441 we know that β α hh We differentiate this equation and use the chain rule 2511 in combination with 2510 For c < t < d we obtain D β t β t D β t h tα ht h tα ht + h td β t α ht h tα ht + h td h tα ht α ht h tα ht + h t 2 D α ht α ht Since this holds for all t we may rewrite the result as Pseudo Riemannian Metrics After having studied 1-forms on a manifold in Section 246 it is natural to investigate covariant tensor fields of degree 2 In this generality not much can be said To make progress one must make a restrictive assumption on the covariant tensor field Definition 2512 Suppose that σ is a covariant tensor field of degree 2 on a differentiable manifold M We say that σ is a symmetric bilinear form provided that 2513 σxy σyx for all X Y XM Similarly σ is called an antisymmetric bilinear form if 2514 σxy σyx for all XY XM We shall concentrate on symmetric bilinear forms that satisfy a nondegeneracy condition that we now explain Notice that any covariant tensor field σ of degree 2 on a manifold M whether or not it is symmetric gives rise to a tensor σ p on each tangent space M p Explicitly σ p is given as follows Let xy M p ; then σ p xy σxyp where XY XM are chosen so that X p x and Y p y Since σ is a tensor field it is linear in both its arguments with respect to functions This fact implies that the definition of σ p does not depend on the choice of X and Y extending X p x and Y p y

9 252 PSEUDO RIEMANNIAN METRICS 855 Definition 2513 A symmetric bilinear form σ on a differentiable manifold M is called nondegenerate provided that σ p is nondegenerate on each tangent space M p for each point p M meaning σ p xy 0 for all y M p x 0 A nondegenerate symmetric bilinear form is called a pseudo Riemannian metric A theorem from linear algebra states that a symmetric bilinear form σ on an n-dimensional vector space V over R can be diagonalized This means that there exists a basis {f 1 f n } of V for which σf i f j 0 whenever i j To understand this fact in a different way we can start with any given basis {e 1 e n } of R n and note that the real numbers σe 1 e j are the entries of a symmetric matrix S An equivalent theorem now asserts that there exists an orthogonal matrix P recall Section 231 for which P 1 SP P T SP is diagonal A similar idea occurs in the proof of Lemma 232 on page 770 and the j th column of P expresses the new basis vector f j in terms of the e i In the case that σ is nondegenerate we can postmultiply P by a diagonal matrix of square roots so that the entries of P T SP become each ±1 It follows that the basis {f 1 f n } can be chosen so that σf i f i ±1 for each i with 1 i n Moreover the number p of plus signs and the number q n p of minus signs does not depend on the choice of the basis {f 1 f n } The pair p q is called the signature of the symmetric bilinear form A particularly important special case occurs when the signature of the symmetric bilinear form is n 0 In this case one says that σ is positive definite Definition 2514 A symmetric bilinear form σ on a differentiable manifold M is called a Riemannian metric if σ p is positive definite on M p for each point p M; equivalently σ p xx > 0 for all nonzero x M p We shall see that any regular surface M in R n acquires such a form σ We shall use the word metric to mean either a pseudo-riemannian metric or a Riemannian metric Usually we denote a metric by and if it is positive definite we write x xx for any tangent vector x to M Beware that when is not positive definite xx can be zero even if x is not

10 856 CHAPTER 25 RIEMANNIAN MANIFOLDS Definition 2515 A pseudo Riemannian manifold is a differentiable manifold M equipped with a metric If xy are tangent vectors to M we call xy the inner product of x and y Similarly we can speak about the inner product of vector fields If XY XM then XY is the function or rather element of FM defined by XY p X p Y p p M A Riemannian manifold is a differentiable manifold M equipped with a positive definite metric Example: The metric on Euclidean space There is a natural positive definite metric on R n given by n 2515 f i g j f i g i u i u j j1 This metric coincides with the standard dot product on R n that we introduced in Section 11 and it is this object that induces the metric defined in Section 121 on surfaces A more general metric on R n is given by 2516 n f i u i j1 g j u j a i f i g i where a 1 a n are nonzero constants The metric 2516 is positive definite if and only if all the a i s are positive We shall make use of 2516 with n 3 and a 1 a 2 a in the next chapter see the explanation of Figure 261 on page 876 An even more general metric is given by 2517 n f i u i j1 g j u j a ij f i g j ij1 where A a ij denotes a symmetric matrix of constants whose determinant is nonzero The metric is positive definite if and only if the eigenvalues of A which are necessarily real are all positve The notion of gradient that we defined on page 274 has a generalization to pseudo-riemannian manifolds Definition 2516 Let θ be a 1-form on a pseudo-riemannian manifold M The dual of θ is the vector field denoted θ defined by θ Y θy for all Y XM

11 252 PSEUDO RIEMANNIAN METRICS 857 Thus θ XM In the special case when θ df for some f FM we call df the gradient of f and denote it by gradf; it is given by gradfy Y[f] That θ is well defined is a consequence of an elementary fact: Lemma 2517 Let V be a vector space with an inner product Any vector x V is completely determined by the mapping V R defined by y x y Proof If xy x y for all y V then x x y 0 for all y V It follows from the nondegeneracy of that x x It is fortunate for the study of pseudo-riemannian manifolds that there is a canonical choice for the covariant derivative Definition 2518 Let M be a pseudo-riemannian manifold The Riemannian connection 2518 of M is defined by : XM XM XM X YZ X YZ + Y XZ Z XY X [YZ] Y [XZ] + Z [XY] for XYZ XM The proof of the following lemma is straightforward: Lemma 2519 A Riemannian connection has the following properties: fx+gy f X + g Y X Y + Z X Y + X Z X fy X[f]Y + f X Y X Y Y X [XY] X YZ X YZ + Y X Z for XYZ XM f g FM and a b R Conversely if a mapping 2518 satisfies then it is given by 2519

12 858 CHAPTER 25 RIEMANNIAN MANIFOLDS Remarks 1 Lemma 2517 implies that X Y is completely determined by The first three properties confirm that really is a covariant derivative They imply that 256 is satisfied with in place of D 3 A general connection D on a differentiable manifold may or may not satisfy condition 2523 which expresses the vanishing of a tensor called the torsion see Exercise 4 A connection that does satisfy 2523 is said to be torsion free It is possible to characterize a Riemannian connection Lemma 2520 Let be a pseudo-riemannian metric on a manifold M Then there is a unique connection on M that satisfies 2523 and 2524 The proof is left as Exercise 5 It is customary to use rather than D to denote what we shall now call the Riemannian connection Often it is called the Levi Civita connection 253 The Classical Treatment of Metrics In the next chapter we shall specialize the general theory of pseudo-riemannian manifolds to the 2-dimensional case We know from Lemma 247 that a regular surface M in R n is a differentiable manifold We can now move on to consider the geometry of surfaces freed from the requirement that they be embedded in R 3 or even R n Definition 2521 A 2-dimensional differentiable manifold equipped with a Riemannian or pseudo-riemannian metric is called an abstract surface The metric on an abstract surface is a generalization of the induced metric that we studied in Chapter 12 Lemma 247 implies: Lemma 2522 A regular surface M equipped with the metric induced from R n is an abstract surface In the literature one frequently sees a metric in n dimensions written as 2525 ds 2 g ij dx i dx j ij1 We gave an intuitive interpretation of 2525 in Chapter 12 in terms of infinitesimals for n 2 To make sense of 2525 in the terminology of this chapter let M be a Riemannian manifold with metric We write ds 2

13 253 CLASSICAL TREATMENT OF METRICS 859 and choose a coordinate system x 1 x n on M This coordinate system gives rise to 1-forms dx 1 dx n The symmetric product of dx i and dx j is the covariant tensor field of degree 2 denoted in one of the equivalent ways dx i dx j dx i dx j dx i dx j and defined using the justaposition notation by dx i dx j XY 1 2 dxi Xdx j Y + dx i Ydx j X X[xi ]Y[x j ] + Y[x i ]X[x j ] 1 2 for XY XM Insertion of the factor 1/2 is to some extent justified by the next lemma Finally let g ij x i x j for 1 i j n The g ij s are called the components of the metric Lemma 2523 A metric ds 2 is given in terms of the g ij s by the formula 2526 ds 2 g ij dx i dx j Proof By definition ds 2 x k ij1 x l x k g kl x l On the other hand we have dx i dx j 1 x i x j + 1 x i x j 1 x k x l 2 x k x l 2 x l x 2 δ ikδ jl + δ il δ jk k so that 2527 ij1 It therefore follows that g ij dx i dx j x k ds 2 x k x l x l ij1 1 2 ij1 g ij δ ik δ jl + δ il δ jk g kl g ij dx i dx j x k for 1 k l n For general XY XM we write X k1 a k and Y x k l1 b l x l x l

14 860 CHAPTER 25 RIEMANNIAN MANIFOLDS Then 2527 implies that ds 2 XY a k b l g ij dx i dx j x k x l kl1 ij1 g ij dx i dx j XY ij1 Since 2528 holds for all XY XM we get 2526 Here is some additional classical terminology Definition 2524 Let g ij 1 i j n be the components of a metric with respect to a coordinate system x 1 x n ; then g ij is the inverse matrix of g ij Frequently the entries g ij are called the upper g ij s while the entries g ij are called the lower g ij s Finally we put 2528 G detg ij Lemma 2525 For any coordinate system x 1 x n on a Riemannian manifold M each 1-form dx i is dual to for i 1 n; that is h1 dx i g ih h1 x h g ih x h Proof By definition of inverse matrix we have δ ij g ih g hj g ih n x h x j h1 On the other hand h1 dx i x j Then Lemma 2517 implies that dx i and h1 dx i δ ij x j g ih x h x j coincide h1 g ih x h

15 254 CHRISTOFFEL SYMBOLS Christoffel Symbols in Riemannian Geometry The Riemannian connection can be used as a means to define Christoffel symbols on a pseudo-riemannian manifold M We shall see in the next chapter that when M is a surface these are essentially the same objects defined in Section 173 However we shall use lower case notation γij k for the time being to avoid possible confusion It is also closer to the notation of the notebooks The Christoffel symbols arise when we attempt to express the Riemannian connection in terms of local coordinates Definition 2526 Let be a metric on an n-dimensional manifold M endowed with a metric ds 2 and let be the Riemannian connection of ds 2 If x 1 x n is a system of local coordinates on M then the Christoffel symbols γ k ij of relative to x 1 x n are given by 2529 for 1 i j k n γ k ij dx k x i x j The next lemma is an easy consequence of this definition and Theorem 2425 It is an effective alternative definition of the Christoffel symbols for Riemannian manifolds and as we shall soon see surfaces Lemma 2527 Let x 1 x n be a coordinate system on a manifold M endowed with a metric ds 2 and let be the Riemannian connection of ds 2 Then γ k 2530 ij x j x k for 1 i j n x i k1 Replacing by D it is of course possible to define the Christoffel symbols for a general connection on a manifold But there is a symmetry property of the Christoffel symbols of a Riemannian connection that may not hold for a general connection Lemma 2528 The Christoffel symbols of a Riemannian connection satisfy the symmetry relation 2531 γij k γji k for i j k 1 n Proof Equation 2531 is an immediate consequence of two facts: i is torsion-free equation 2523 and ii the brackets of coordinate vector fields vanish Corollary 2458 on page 836

16 862 CHAPTER 25 RIEMANNIAN MANIFOLDS Next we derive the classical expression for the γ i jk s in terms of the g ij and the g ij In practice the following result provides the most efficient way of computing the Christoffel symbols and it is the basis for a program in Notebook 25 Lemma 2529 The Christoffel symbols γjk i of a Riemannian connection are given by γjk i 1 g ih ghj + g hk g jk x k x j x h h1 Proof Since brackets of coordinate vector fields are zero 2519 reduces to 2 + x j x k x h x j x k x h x k x j x h x h x j x k g kh x j + g jh x k g jk x h It follows from the definition of γjk i and Lemma 2525 that γjk i dx i x j x dx i k x j x k The result now follows Finally we generalize 2530 h1 g ih x j x k Lemma 2530 Let XY XM and let x 1 x n be a coordinate system on a Riemannian manifold M Then 2533 X Y XY[x k ] + X[x i ]Y[x j ]γij k x k Proof We have k1 X Y X[xi] x i ij1 ij1 Y[x j ] x j j1 X[x i ] Y[x j ] + Y[x j ] x i x j x i x j XY[x j ] + X[x i ]Y[x j ]γ k ij x j x k ijk1 XY[x k] + X[x i ]Y[x j ]γij k x k j1 k1 ij1 The Christoffel symbols are therefore used to modify the notion of differentiation that a patch provides so as to obtain a patch-independent derivative x h

17 255 RIEMANN CURVATURE TENSOR The Riemann Curvature Tensor According to Lemma 2527 the Christoffel symbols are the components of the mapping : XM XM XM defined by XY X Y We know from the definition of a covariant derivative that this mapping is not a vectorvariant tensor because it is not linear over XM see page 836 For this reason a change of chart results in a rather complicated change of the symbols γjk i which we describe next Let x: U M and x y: V M be two charts with corresponding systems of coordinates x i and y i Then we may write 2534 x k q1 y q x k y q k 1 n where J y i /x j denotes the Jacobian matrix of the change of patch The situation is essentially that described on page 828 and 2534 corresponds to 2413 in which Φ is the identity mapping In this section we shall however blur the distinction between functions on the open set W xu yv of M and functions on the open sets x 1 W and y 1 W of R n We may now repeat the proof of Lemma 2530 by taking XY to be vector fields arising from the first chart From 2534 we get 2535 x j x k 2 y q q1 q1 x j x k 2 y q + x j x k y q Using Lemma 2527 this can be expressed as γjk i 2 y r + x i x j x k r1 y q + y q x k x j pq1 pq1 y q y p x j y q x k y p x j y q x k γ r pq yp y r y q where γ pq r are Christoffel symbols associated to the patch y A final application of 2534 yields 2536 γjk i y r 2 y r + x i x j x k pq1 y p x j y q x k γ r pq for r 1 n where γ r pq are the Christoffel symbols relative to y It is the presence of the first term on the right-hand side violates the transformation rule for tensors

18 864 CHAPTER 25 RIEMANNIAN MANIFOLDS It is a remarkable fact that one may nonetheless take derivatives of the Christoffel symbols and combine them with quadratic terms in the same symbols so as to obtain quantities that transform more simply than 2536 The book [McCl] contains a good exposition of the calculations that led Riemann to discover the curvature tensor defined next Definition 2531 The Riemann curvature tensor is the mapping defined 3 by R: XM XM XM XM 2537 RXYZ X Y Z Y X Z [XY] Z The Lie bracket [XY] of vector fields is defined on page 832 The fact that R is a tensor field is not obvious but can easily be verified see Exercise 6 There are a number of tensors associated to R all of which define essentially the same object For example for each XY we may define a mapping R XY : XM XM by R XY Z RXYZ Abbreviating the left-hand side we may replace RXYZ by R XY Z which as we are about to see is a more useful expression Furthermore we may manufacture a covariant tensor field by combining 2537 with the metric: RXYZW RXYZW R XY ZW The use of the same symbol R in these slightly different contexts does not cause confusion Lemma 2532 The Riemann tensor field has the following symmetries: R XY R YX R XY ZW R XY WZ R XY ZW R ZW XY R XY Z + R YZ X + R ZX Y 0 3 Approximately 40% of books on Riemannian manifolds define the curvature to be the negative of the one given here but this sign will pop up again in Lemma 2534

19 255 RIEMANN CURVATURE TENSOR 865 Equation 2541 is called the first Bianchi identity Algebraically the curvature tensor is a rather complicated object and often it is important to study its so-called contractions There are two such contractions; to define them we choose a set of vector fields {E 1 E n } on an open set of M which are orthonormal at each point Definition 2533 The Ricci and scalar curvatures of a Riemannian manifold are defined by 2542 ρxy R XEm E m Y and 2543 s m1 ρe l E l l1 R El E m E m E l lm1 It is inherent in this definition that the quantities ρ and s are independent of the choice of orthonormal basis The reason for this will become clearer after we have discussed coordinate descriptions The scalar curvature is merely a function and its value at a given point of M is a number which is an invariant of the Riemannian structure The Ricci tensor ρ is a symmetric bilinear form see Definition 2512 and Exercise 8; its value at each point is determined by a symmetric n n matrix or at most nn + 1/2 numbers though only 1 is necessary in dimension 2 The Ricci tensor is therefore exactly the same sort of tensor as the metric itself This fact allows one to combine the Ricci tensor and the metric in the same equation and Einstein s equations of general relativity do just that see [HaEl chapter 3] n R ρ s The nature of the Riemann tensor is less obvious but by analysing the symmetries in dimension n it can be shown that R is determined by a total of n 2 n 2 1/12 numbers at each point see for example [Spivak]

20 866 CHAPTER 25 RIEMANNIAN MANIFOLDS The table above displays this numerical information for n 7 and shows that in general the Ricci tensor accounts for a relatively small part of the full curvature tensor R Nonetheless it is of vital importance in lower dimensions When n 2 R is completely determined by s or as we shall see the Gaussian curvature K When n 3 the Ricci tensor ρ determines all of R; when n 4 it determines exactly half of R Pick a chart x: U M with associated coordinates x 1 x n and set R ijkl R x k x l x j x i The strange order of the indices is to retain consistency with the definition of the related scalar quantities R i jkl adopted from [HaEl chapter 2] by means of the formula The two are related by R x k x l x j R i jkl n x i R ijkl n g im R m jkl or R i jkl n g im R mjkl m1 This sort of index shifting is common in tensor calculus and we have already seen it in Lemma 2525 The metric can also be used to perform contractions; with our conventions the Ricci and scalar curvatures are given by 2544 ρ jl ρ x j s n g jl ρ jl jl1 n R i jil x l n ijkl1 n ik1 m1 g jl g ik R ijkl g ik R ijkl The curvature tensor itself is given in local coordinates by Lemma 2534 With the notation above 2545 Proof R i jkl γi jk γi jl + n γ m x l x jk γlm i γm jl γi km k m1 Because the brackets between the vector fields /x i vanish we obtain R x k x l x j x k x j x l x j x l n x k m1 γ m lj x k n x m x l m1 The result follows from 2531 and steps of the type 2535 γ m kj x m When we expand 2545 for n 2 we obtain formulas that resemble the right-hand side of equations 1911 on page 601 We shall be more explicit in the next chapter once we have identified the Christoffel symbols γjk i with those defined for a surface in Chapter 17

21 256 EXERCISES Exercises 1 Complete the proof of Lemma Give examples of vector fields XYZ on R such that Y 0 Z 0 but D X Y 0 D X Z 0 3 Prove Lemmas 2519 and The torsion of a connection D on a differentiable manifold M is defined by TXY D X Y D Y X [XY] Show that T is a vectorvariant tensor field on M that satisfies refer to page 836 TYX TXY 5 Prove Lemma 2520 by simplifying the right-hand side of Verify that the curvature tensor as defined in 2537 on page 864 is a vectorvariant tensor field 7 Deduce from the previous exercise that if x and x y are two charts with respective curvature components R ijkl R pqrs then 2546 R ijkl pqrs1 y p y q y r y s R pqrs x i x j x k x l 8 Show that the Ricci tensor satisfies ρxy ρyx and that for n 2 ρxy 1 2 s XY See also Theorem 264 on page 874 of the next chapter