Chapter 14
Curvature in Riemannian Manifolds
Since the notion of curvature can be defined for curves and surfaces, it is natural to wonder whether it can be generalized to manifolds of dimension n 3. Such a generalization does exist and was first proposed by Riemann.
However, Riemann’s seminal paper published in 1868 two years after his death only introduced the sectional cur- vature, and did not contain any proofs or any general methods for computing the sectional curvature.
659 660 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Fifty years or so later, the idea emerged that the cur- vature of a Riemannian manifold M should be viewed as a measure R(X, Y )Z of the extent to which the op- erator (X, Y ) X Y Z is symmetric, where is a connection on M7!(where r r X, Y, Z are vector fields,r with Z fixed).
It turns out that the operator R(X, Y )Z is C1(M)- linear in all of its three arguments, so for all p M, it defines a trilinear map 2
R : T M T M T M T M. p p ⇥ p ⇥ p ! p The curvature operator R is a rather complicated object, so it is natural to seek a simpler object. 14.1. THE CURVATURE TENSOR 661 Fortunately, there is a simpler object, namely the sec- tional curvature K(u, v), which arises from R through the formula
K(u, v)= R(u, v)u, v , h i for linearly independent unit vectors u, v.
When is the Levi-Civita connection induced by a Rie- mannianr metric on M,itturnsoutthatthecurvature operator R can be recovered from the sectional curva- ture.
Another important notion of curvature is the Ricci cur- vature,Ric(x, y), which arises as the trace of the linear map v R(x, v)y. 7! As we said above, if M is a Riemannian manifold and if is a connection on M,theRiemanniancurvature R(rX, Y )Z measures the extent to which the operator (X, Y ) Z is symmetric (for any fixed Z). 7! rXrY 662 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS If (M, , )isaRiemannianmanifoldofdimensionn, and ifh the i connection on M is the flat connection, which means that r
@ =0,i=1,...,n, rX @x ✓ i◆ for every chart (U,')andallX X(U), since every vector field Y on U can be written2 uniquely as
n @ Y = Y i@x i=1 i X for some smooth functions Yi on U,foreveryothervector field X on U,becausetheconnectionisflatandbythe Leibniz property of connections, we have
@ @ @ @ Y = X(Y ) + Y = X(Y ) . rX i@x i @x irX @x i @x ✓ i◆ i ✓ i◆ i 14.1. THE CURVATURE TENSOR 663 Then it is easy to check that the above implies that
Z Z = Z, rXrY rY rX r[X,Y ] for all X, Y, Z X(M). Consequently, it is natural to define the deviation2 of a connection from the flat connec- tion by the quantity
R(X, Y )Z = Z Z Z rXrY rY rX r[X,Y ] for all X, Y, Z X(M). 2 Definition 14.1. Let (M,g)beaRiemannianmanifold, and let be any connection on M.Theformula r R(X, Y )Z = Z Z Z, rXrY rY rX r[X,Y ] for X, Y, Z X(M), defines a function 2 R: X(M) X(M) X(M) X(M) ⇥ ⇥ ! called the Riemannian curvature of M. 664 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Proposition 14.1. Let M be a manifold with any connection . The function r R: X(M) X(M) X(M) X(M) ⇥ ⇥ ! given by
R(X, Y )Z = Z Z Z rXrY rY rX r[X,Y ] is C1(M)-linear in X, Y, Z, and skew-symmetric in X and Y . As a consequence, for any p M, 2 (R(X, Y )Z)p depends only on X(p),Y(p),Z(p). 14.1. THE CURVATURE TENSOR 665 It follows that R defines for every p M atrilinearmap 2 R : T M T M T M T M. p p ⇥ p ⇥ p ! p Experience shows that it is useful to consider the family of quadrilinear forms (unfortunately!) also denoted R, given by
R (x, y, z, w)= R (x, y)z,w , p h p ip as well as the expression R(x, y, y, x), which, for an or- thonormal 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. 666 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS 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,thelatterchoice corresponds to using R(x, y)insteadofR(x, y), that is, to define R(X, Y )Z by
R(X, Y )Z = Z + Z Z. r[X,Y ] rY rX rXrY
As pointed out by Milnor [36] (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 h h i j ki coincides with the classical Ricci notation, Rhijk.
Gallot, Hulin and Lafontaine [22] (Chapter 3, Section A.1) give other reasons supporting this choice of sign. 14.1. THE CURVATURE TENSOR 667 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 14.2. Let (M, , )beaRiemannianman- ifold equipped with the Levi-Civitah i connection. The cur- vature tensor is the family of trilinear functions R : T M T M T M T M defined by p p ⇥ p ⇥ p ! p
R (x, y)z = Z + Z Z, p r[X,Y ] rY rX rXrY for every p M and for any vector fields X, Y, Z 2 2 X(M)suchthatx = X(p), y = Y (p), and z = Z(p). 668 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS The family of quadrilinear forms associated with R,also denoted R,isgivenby
R (x, y, z, w)= (R (x, y)z,w , p h p i for all p M and all x, y, z, w T M. 2 2 p Locally in a chart, we write
@ @ @ l @ R , = Rjhi @xh @xi @xj @xl ✓ ◆ Xl and
@ @ @ @ l Rhijk = R , , = glkRjhi. @xh @xi @xj @xk ⌧ ✓ ◆ Xl 14.1. THE CURVATURE TENSOR 669
l The coe cients Rjhi can be expressed in terms of the k Christo↵el symbols ij,byaratherunfriendlyformula (see Gallot, Hulin and Lafontaine [22] (Chapter 3, Section 3.A.3) or O’Neill [43] (Chapter III, Lemma 38).
Since we have adopted O’Neill’s conventions for the order l of the subscripts in Rjhi,hereistheformulafromO’Neill:
Rl = @ l @ l + l m l m. jhi i hj h ij im hj hm ij m m X X There is another way of defining the curvature tensor which is useful for comparing second covariant derivatives of one-forms.
For any fixed vector field Z,themapY Z from 7! rY X(M)to (M)isaC1(M)-linear map that we will denote XZ (this is a (1, 1) tensor). r 670 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS
The covariant derivative X Z of Z is defined by r r r
( X( Z))(Y )= X( Y Z) ( Y )Z. r r r r rrX
2 Usually, ( X( Z))(Y )isdenotedby X,Y Z,and r r r 2 X,Y Z = X( Y Z) Y Z r r r rrX is called the second covariant derivative of Z with re- spect to X and Y .
Proposition 14.2. Given a Riemanniain manifold (M,g), if is the Levi-Civita connection induced by g, then ther curvature tensor is given by R(X, Y )Z = 2 Z 2 Z. rY,X rX,Y
We already know that the curvature tensor has some sym- metry 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 proposi- tion. 14.1. THE CURVATURE TENSOR 671 Proposition 14.3. For a Riemannian manifold (M, , ) equipped with the Levi-Civita connection, the curvatureh i 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 next proposition will be needed in the proof of the second variation formula. Recall the notion of a vector field along a surface given in Definition 13.9. 672 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Proposition 14.4. For a Riemannian manifold (M, , ) equipped with the Levi-Civita connection, h i for every parametrized surface ↵: R2 M, for every ! vector field V X(M) along ↵, we have 2 D D D D @↵ @↵ V V = R , V. @y @x @x @y @x @y ✓ ◆ Remark: Since the Levi-Civita connection is torsion- free, it is easy to check that D @↵ D @↵ = . @x @y @y @x
The curvature tensor is a rather complicated object.
Thus, it is quite natural to seek simpler notions of curva- ture.
The sectional curvature is indeed a simpler object, and it turns out that the curvature tensor can be recovered from it. 14.2. SECTIONAL CURVATURE 673 14.2 Sectional Curvature
Basically, the sectional curvature is the curvature of two- dimensional sections of our manifold.
Given any two vectors u, v TpM,recallbyCauchy- Schwarz that 2 u, v 2 u, u v, v , h ip h iph ip with equality i↵ u and v are linearly dependent.
Consequently, if u and v are linearly independent, we have
u, u v, v u, v 2 =0. h iph ip h ip 6 In this case, we claim that the ratio
R (u, v, u, v) K(u, v)= p u, u v, v u, v 2 h iph ip h ip is independent of the plane ⇧spanned by u and v. 674 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Definition 14.3. Let (M, , )beanyRiemannian manifold equipped with theh Levi-Civita i connection. For every p TpM,forevery2-plane⇧ TpM,thesec- tional curvature2 K(⇧)of⇧is given by✓
R (x, y, x, y) K(⇧) = K(x, y)= p , x, x y, y x, y 2 h iph ip h ip for any basis (x, y)of⇧.
Observe that if (x, y)isanorthonormalbasis,thenthe denominator is equal to 1.
The expression Rp(x, y, x, y)isoftendenotedp(x, y).
Remarkably, p determines Rp.Wedenotethefunction p by . 7! p 14.2. SECTIONAL CURVATURE 675 Proposition 14.5. Let (M, , ) be any Rieman- nian manifold equipped withh the i Levi-Civita connec- tion. The function determines the curvature tensor R. Thus, the knowledge of all the sectional curvatures determines the curvature tensor. Moreover, we have
6 R(x, y)z,w = (x + w, y + z) (x, y + z) h i (w, y + z) (y + w, x + z) +(y, x + z)+(w, x + z) (x + w, y)+(x, y)+(w, y) (x + w, z)+(x, z)+(w, z) + (y + w, x) (y, x) (w, x) + (y + w, z) (y, z) (w, z).
For a proof of this formidable equation, see Kuhnel [30] (Chapter 6, Theorem 6.5).
Adi↵erentproofoftheaboveproposition(withoutan explicit formula) is also given in O’Neill [43] (Chapter III, Corollary 42). 676 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Let R (x, y)z = x, z y y, z x. 1 h i h i Observe that R (x, y)x, y = x, x y, y x, y 2. h 1 i h ih i h i As a corollary of Proposition 14.5, we get:
Proposition 14.6. Let (M, , ) be any Rieman- nian manifold equipped withh the i Levi-Civita connec- tion. If the sectional curvature K(⇧) does not depend on the plane ⇧ but only on p M, in the sense that 2 K is a scalar function K : M R, then ! R = KR1.
In particular, in dimension n =2,theassumptionof Proposition 14.6 holds and K is the well-known Gaussian curvature for surfaces. 14.2. SECTIONAL CURVATURE 677 Definition 14.4. ARiemannianmanifold(M, , ) is said to have constant (resp. negative, resp.h pos- i itive) curvature i↵its sectional curvature is constant (resp. negative, resp. positive).
In dimension n 3, we have the following somewhat surprising theorem due to F. Schur:
Proposition 14.7. (F. Schur, 1886) Let (M, , ) be a connected Riemannian manifold. If dim(Mh ) i3 and if the sectional curvature K(⇧) does not depend on the plane ⇧ TpM but only on the point p M, then K is constant✓ (i.e., does not depend on p).2
The proof, which is quite beautiful, can be found in Kuh- nel [30] (Chapter 6, Theorem 6.7). 678 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS If we replace the metric g = , by the metric g = , where >0isaconstant,somesimplecalcu-h i h i lations show that the Christo↵el symbols and the Levi-e Civita connection are unchanged, as well as the curvature tensor, but the sectional curvature is changed, with 1 K = K.
As a consequence, if Meis a Riemannian manifold of con- stant curvature, by rescaling the metric, we may assume that either K = 1, or K =0,orK =+1. Here are standard examples of spaces with constant cur- vature.
(1) The sphere Sn Rn+1 with the metric induced by ✓ Rn+1,where
n n+1 2 2 S = (x1,...,xn+1) R x + + x =1 . { 2 | 1 ··· n+1 }
The sphere Sn has constant sectional curvature K = +1. 14.2. SECTIONAL CURVATURE 679
(2) Euclidean space Rn+1 with its natural Euclidean met- ric. Of course, K =0. + (3) The hyperbolic space n (1) from Definition 6.1. Re- call that this space is definedH in terms of the Lorentz n+1 innner product , 1 on R ,givenby h i
n+1 (x ,...,x ), (y ,...,y ) = x y + x y . h 1 n+1 1 n+1 i1 1 1 i i i=2 X
By definition, +(1), written simply Hn,isgivenby Hn
n n+1 H = x =(x1,...,xn+1) R { 2 x, x = 1,x > 0 . |h i1 1 }
n It can be shown that the restriction of , 1 to H is positive, definite, which means thath it is i a metric n on TpH . 680 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS n The space H equipped with this metric gH is called hyperbolic space and it has constant curvature K = 1.
There are other isometric models of Hn that are perhaps intuitively easier to grasp but for which the metric is more complicated.
For example, there is a map PD: Bn Hn where Bn = ! x Rn x < 1 is the open unit ball in Rn,givenby { 2 |k k } 1+ x 2 2x PD(x)= k k2, 2 . 1 x 1 x ! k k k k
It is easy to check that PD(x), PD(x) 1 = 1andthat PD is bijective and an isometry.h i 14.2. SECTIONAL CURVATURE 681
One also checks that the pull-back metric gPD =PD⇤gH on Bn is given by
4 2 2 gPD = (dx + + dx ). (1 x 2)2 1 ··· n k k
The metric gPD is called the conformal disc metric,and n the Riemannian manifold (B ,gPD)iscalledthePoincar´e disc model or conformal disc model.
The metric gPD is proportional to the Euclidean metric, and thus angles are preserved under the map PD. 682 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Another model is the Poincar´ehalf-plane model n x R x1 > 0 ,withthemetric { 2 | } 1 2 2 gPH = 2 (dx1 + + dxn). x1 ···
We already encountered this space for n =2.
In general, it is practically impossible to find an explicit formula for the sectional curvature of a Riemannian man- ifold.
The spaces Sn, Rn+1,andHn are exceptions.
Nice formulae can be given for Lie groups with bi-invariant metrics (see Chapter 18) and for certain kinds of reductive homogeneous manifolds (see Chapter 20). 14.3. RICCI CURVATURE 683 14.3 Ricci Curvature
The Ricci tensor is another important notion of curvature.
It is mathematically simpler than the sectional curvature (since it is symmetric) but it plays an important role in the theory of gravitation as it occurs in the Einstein field equations.
Recall that if f : E E is a linear map from a finite- dimensional Euclidean! vector space to itself, given any orthonormal basis (e1,...,en), we have n tr(f)= f(e ),e . h i ii i=1 X 684 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Definition 14.5. Let (M, , )beaRiemannianman- ifold (equipped with the Levi-Civitah i connection). The Ricci curvature Ric of M is defined as follows: For ev- ery p M,forallx, y TpM,setRicp(x, y)tobethe trace of2 the endomorphism2 v R (x, v)y.Withrespect 7! p to any orthonormal basis (e1,...,en)ofTpM,wehave
n n Ric (x, y)= R (x, e )y, e = R (x, e ,y,e ). p h p j jip p j j j=1 j=1 X X
The scalar curvature S of M is the trace of the Ricci curvature; that is, for every p M, 2
S(p)= R(ei,ej,ei,ej)= K(ei,ej), i=j i=j X6 X6 where K(ei,ej)denotesthesectionalcurvatureofthe plane spanned by ei,ej. 14.3. RICCI CURVATURE 685 In a chart the Ricci curvature is given by
@ @ R =Ric , = Rm , ij @x @x ijm i j m ✓ ◆ X and the scalar curvature is given by
ij S(p)= g Rij, i,j X where (gij)istheinverseoftheRiemannmetricmatrix (gij). See O’Neill, pp. 87-88 [43]
If M is equipped with the Levi-Civita connection, in view of Proposition 14.3 (4), the Ricci curvature is symmetric. 686 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS The tensor Ric is a (0, 2)-tensor but it can be interpreted as a (1, 1)-tensor as follows.
Definition 14.6. Let (M, , )beaRiemannianman- ifold (equipped with any connection).h i The (1, 1)-tensor # Ricp is defined to be
Ric#u, v =Ric(u, v), h p ip p for all u, v T M. 2 p
Proposition 14.8. Let (M,g) be a Riemannian man- ifold and let be any connection on M.If(e ,...,e ) r 1 n is any orthonormal basis of TpM, we have n # Ricp (u)= Rp(ej,u)ej. j=1 X 14.3. RICCI CURVATURE 687 Then it is easy to see that
# S(p)=tr(Ricp ).
This is why we said (by abuse of language) that S is the trace of Ric.
Observe that in dimension n =2,wegetS(p)=2K(p). Therefore, in dimension 2, the scalar curvature deter- mines the curvature tensor.
In dimension n =3,itturnsoutthattheRiccitensor completely determines the curvature tensor, although this is not obvious.
Since Ric(x, y)issymmetric,Ric(x, x)determinesRic(x, y) completely. 688 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS
Observe that for any orthonormal frame (e1,...,en)of TpM,usingthedefinitionofthesectionalcurvatureK, we have
n n Ric(e ,e )= (R(e ,e)e ,e = K(e ,e). 1 1 h 1 i 1 ii 1 i i=1 i=2 X X
Thus, Ric(e1,e1)isthesumofthesectionalcurvatures of any n 1orthogonalplanesorthogonaltoe1 (a unit vector).
For a Riemannian manifold with constant sectional cur- vature, we have
Ric(x, x)=(n 1)Kg(x, x),S= n(n 1)K, where g = , is the metric on M. h i 14.3. RICCI CURVATURE 689 Spaces for which the Ricci tensor is proportional to the metric are called Einstein spaces.
Definition 14.7. ARiemannianmanifold(M,g)iscalled an Einstein space i↵the Ricci curvature is proportional to the metric g;thatis:
Ric(x, y)= g(x, y), for some function : M R. ! If M is an Einstein space, observe that S = n . 690 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Remark: For any Riemanian manifold (M,g), the quan- tity S G =Ric g 2 is called the Einstein tensor (or Einstein gravitation tensor for space-times spaces).
The Einstein tensor plays an important role in the theory of general relativity. For more on this topic, see Kuhnel [30] (Chapters 6 and 8) O’Neill [43] (Chapter 12). 14.4. THE SECOND VARIATION FORMULA AND THE INDEX FORM 691 14.4 The Second Variation Formula and the Index Form
In Section 13.6, we discovered that the geodesics are ex- actly the critical paths of the energy functional (Theorem 13.24).
For this, we derived the First Variation Formula (Theo- rem 13.23).
It is not too surprising that a deeper understanding is achieved by investigating the second derivative of the en- ergy functional at a critical path (a geodesic).
By analogy with the Hessian of a real-valued function on Rn,itispossibletodefineabilinearfunctional
I : T ⌦(p, q) T ⌦(p, q) R ⇥ ! when is a critical point of the energy function E (that is, is a geodesic). 692 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS This bilinear form is usually called the index form.
Note that Milnor denotes I by E and refers to it as the ⇤⇤ Hessian of E,butthisisabitconfusingsinceI is only defined for critical points, whereas the Hessian is defined for all points, critical or not.
Now, if f : M R is a real-valued function on a finite- dimensional manifold! M and if p is a critical point of f, which means that df p =0,itisnothardtoprovethat there is a symmetric bilinear map I : TpM TpM R such that ⇥ !
I(X(p),Y(p)) = Xp(Yf)=Yp(Xf), for all vector fields X, Y X(M). 2 14.4. THE SECOND VARIATION FORMULA AND THE INDEX FORM 693 Furthermore, I(u, v)canbecomputedasfollows:forany 2 u, v TpM,foranysmoothmap↵: R R such that 2 ! @↵ @↵ ↵(0, 0) = p, (0, 0) = u, (0, 0) = v, @x @y we have
@2(f ↵)(x, y) I(u, v)= . @x@y (0,0) 694 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS The above suggests that in order to define
I : T ⌦(p, q) T ⌦(p, q) R, ⇥ ! that is to define I (W ,W ), where W ,W T ⌦(p, q) 1 2 1 2 2 are vector fields along (with W1(0) = W2(0) = 0 and W1(1) = W2(1) = 0), we consider 2-parameter varia- tions
↵: U [0, 1] M, ⇥ ! where U is an open subset of R2 with (0, 0) U,such that 2
@↵ ↵(0, 0,t)= (t), (0, 0,t)=W1(t), @u1 @↵ (0, 0,t)=W2(t). @u2 See Figure 14.1. 14.4. THE SECOND VARIATION FORMULA AND THE INDEX FORM 695
q
p
α
U x [0, 1] u 2
u 1
Figure 14.1: A 2-parameter variation ↵.Thepinkcurvewithitsassociatedvelocityfieldis ↵(0, 0,t)= (t). The blue vector field is W1(t)whilethegreenvectorfieldisW2(t). 696 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Then, we set
@2(E ↵)(u ,u ) I (W ,W )= 1 2 , 1 2 @u @u 1 2 (0,0) e where ↵ ⌦(p, q)isthepathgivenby 2
e ↵(u1,u2)(t)=↵(u1,u2,t).
For simplicity ofe notation, the above derivative if often 2 written as @ E (0, 0). @u1@u2
To prove that I (W1,W2)isactuallywell-defined,we need the following result: 14.4. THE SECOND VARIATION FORMULA AND THE INDEX FORM 697 Theorem 14.9. (Second Variation Formula) Let ↵: U [0, 1] M be a 2-parameter variation of a geodesic⇥ ⌦!(p, q), with variation vector fields W ,W T2⌦(p, q) given by 1 2 2 @↵ @↵ W1(t)= (0, 0,t),W2(t)= (0, 0,t). @u1 @u2 Then, we have the formula
1 @2(E ↵)(u ,u ) dW 1 2 = W (t), 1 2 @u @u 2 t dt 1 2 (0,0) t ⌧ e 1 2X D W1 W2, + R(V,W1)V dt, dt2 Z0 ⌧ where V (t)= 0(t) is the velocity field,
dW1 dW1 dW1 t = (t+) (t ) dt dt dt
dW1 is the jump in dt at one of its finitely many points of discontinuity in (0, 1), and E is the energy function on ⌦(p, q). 698 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS Theorem 14.9 shows that the expression
@2(E ↵)(u ,u ) 1 2 @u @u 1 2 (0,0) e only depends on the variation fields W1 and W2,andthus I (W1,W2)isactuallywell-defined.Ifnoconfusionarises, we write I(W1,W2)forI (W1,W2).
Proposition 14.10. Given any geodesic ⌦(p, q), 2 the map I : T ⌦(p, q) T ⌦(p, q) R defined so that for all W ,W T ⌦(⇥p, q), ! 1 2 2 @2(E ↵)(u ,u ) I(W ,W )= 1 2 , 1 2 @u @u 1 2 (0,0) e only depends on W1 and W2 and is bilinear and sym- metric, where ↵: U [0, 1] M is any 2-parameter variation, with ⇥ !
@↵ ↵(0, 0,t)= (t), (0, 0,t)=W1(t), @u1 @↵ (0, 0,t)=W2(t). @u2 14.4. THE SECOND VARIATION FORMULA AND THE INDEX FORM 699 On the diagonal, I(W, W )canbedescribedintermsof a1-parametervariationof .Infact,
d2E(↵) I(W, W )= (0), du2 e where ↵:( ✏,✏) ⌦(p, q)denotesanyvariationof ! d↵ with variation vector field du (0) equal to W . e e Proposition 14.11. If ⌦(p, q) is a minimal geodesic, then the bilinear2 index form I is positive semi-definite, which means that I(W, W ) 0 for all W T ⌦(p, q). 2 700 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS 14.5 Jacobi Fields and Conjugate Points
Jacobi fields arise naturally when considering the expres- sion involved under the integral sign in the Second Varia- tion Formula and also when considering the derivative of the exponential.
If B : E E R is a symmetric bilinear form defined on some⇥ vector! space E (possibly infinite dimentional), recall that the nullspace of B is the subset null(B)ofE given by
null(B)= u E B(u, v)=0, for all v E . { 2 | 2 }
The nullity ⌫ of B is the dimension of its nullspace.
The bilinear form B is nondegenerate i↵null(B)=(0) i↵ ⌫ =0. 14.5. JACOBI FIELDS AND CONJUGATE POINTS 701 If U is a subset of E,wesaythatB is positive definite (resp. negative definite) on U i↵ B(u, u) > 0(resp. B(u, u) < 0) for all u U,withu =0. 2 6 The index of B is the maximum dimension of a subspace of E on which B is negative definite.
We will determine the nullspace of the symmetric bilinear form I : T ⌦(p, q) T ⌦(p, q) R, ⇥ ! where is a geodesic from p to q in some Riemannian manifold M. 702 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS
Now, if W is a vector field in T ⌦(p, q)andW satisfies the equation
D2W + R(V,W)V =0, ( ) dt2 ⇤ where V (t)= 0(t)isthevelocityfieldofthegeodesic , since W is smooth along ,itisobviousfromtheSecond Variation Formula that
I(W, W )=0, for all W T ⌦(p, q). 2 2 2 Therefore, any vector field in the nullspace of I must satisfy equation ( ). Such vector fields are called Jacobi fields. ⇤ 14.5. JACOBI FIELDS AND CONJUGATE POINTS 703 Definition 14.8. Given a geodesic ⌦(p, q), a vector field J along is a Jacobi field i↵it satisfies2 the Jacobi di↵erential equation
D2J + R( 0,J) 0 =0. dt2
The equation of Definition 14.8 is a linear second-order di↵erential equation that can be transformed into a more familiar form by picking some orthonormal parallel vector fields X1,...,Xn along .
To do this, pick any orthonormal basis (e1,...,en)in T M,withe = 0(0)/ 0(0) ,anduseparalleltransport p 1 k k along to get X1,...,Xn. 704 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS n Then, we can write J = i=1 yiXi,forsomesmooth functions yi,andtheJacobiequationbecomesthesystem of second-order linear ODE’sP
2 n d yi + R( 0,E , 0,E)y =0, 1 i n. dt2 j i j j=1 X
By the existence and uniqueness theorem for ODE’s, for every pair of vectors u, v TpM,thereisauniqueJacobi 2 DJ fields J so that J(0) = u and dt (0) = v.
Since TpM has dimension n,itfollowsthatthedimension of the space of Jacobi fields along is 2n.
DJ Proposition 14.12. If J(0) and dt (0) are orthogonal to 0(0), then J(t) is orthogonal to 0(t) for all t [0, 1]. 2
Proposition 14.13. If J is orthogonal to , which means that J(t) is orthogonal to 0(t) for all t [0, 1], DJ 2 then dt is also orthogonal to . 14.5. JACOBI FIELDS AND CONJUGATE POINTS 705 Proposition 14.14. If ⌦(p, q) is a geodesic in a Riemannian manifold of2 dimension n, then the fol- lowing properties hold:
(1) For all u, v TpM, there is a unique Jacobi fields 2 DJ J so that J(0) = u and dt (0) = v. Consequently, the vector space of Jacobi fields has dimension n. (2) The subspace of Jacobi fields orthogonal to has dimension 2n 2. The vector fields 0 and t 7! t 0(t) are Jacobi fields that form a basis of the sub- space of Jacobi fields parallel to (that is, such that J(t) is collinear with 0(t), for all t [0, 1].) See Figure 14.2. 2 (3) If J is a Jacobi field, then J is orthogonal to i↵ there exist a, b [0, 1], with a = b, so that J(a) and J(b) are both orthogonal2 to 6 i↵there is some a DJ 2 [0, 1] so that J(a) and dt (a) are both orthogonal to . (4) For any two Jacobi fields X, Y along , the expres- sion X, Y Y,X is a constant, and if X hr 0 i hr 0 i and Y vanish at some point on , then 0X, Y Y,X =0. hr i hr 0 i 706 CHAPTER 14. CURVATURE IN RIEMANNIAN MANIFOLDS
q
γ
p
M transparent view of M
q
γ p enlargement of with frame
key X1 X2
X 3 J
Figure 14.2: An orthogonal Jacobi field J for a three dimensional manifold M. Note that J is in the plane spanned by X2 and X3,whileX1 is in the direction of the velocity field. 14.5. JACOBI FIELDS AND CONJUGATE POINTS 707 Following Milnor, we will show that the Jacobi fields in T ⌦(p, q)areexactlythevectorfieldsinthenullspaceof the index form I.
First, we define the important notion of conjugate points.
Definition 14.9. Let ⌦(p, q)beageodesic.Two distinct parameter values 2a, b [0, 1] with a