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Home , Isometry, Jacobi field, Killing vector field

Chapter 16

Isometries, Local , Riemannian Coverings and Submersions, Killing Vector Fields

16.1 Isometries and Local Isometries

Recall that a local between two Riemannian M and N is a smooth ': M N so that !

(d') (u), (d' )(v) = u, v , h p p i'(p) h ip for all p M and all u, v TpM.Anisometry is a local isometry2 and a di↵eomorphism.2

By the inverse theorem, if ': M N is a local isometry, then for every p M,thereissomeopensubset! 2 U M with p U so that ' U is an isometry between U ✓and '(U). 2

743 744 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS Also recall that if ': M N is a di↵eomorphism, then for any vector field X on!M,thevectorfield' X on N (called the push-forward of X)isgivenby ⇤

1 (' X)q = d'' 1(q)X(' (q)), for all q N, ⇤ 2 or equivalently, by

(' X)'(p) = d'pX(p), for all p M. ⇤ 2

For any smooth function h: N R,foranyq N,we have ! 2

X (h)q = X(h ')' 1(q), ⇤ or

X (h)'(p) = X(h ')p. ⇤ 16.1. ISOMETRIES AND LOCAL ISOMETRIES 745 It is natural to expect that isometries preserve all “nat- ural” Riemannian concepts and this is indeed the case. We begin with the Levi-Civita connection.

Proposition 16.1. If ': M N is an isometry, then !

' ( XY )= ' X(' Y ), for all X, Y X(M), ⇤ r r ⇤ ⇤ 2 where XY is the Levi-Civita connection induced by the metricr on M and similarly on N.

As a corollary of Proposition 16.1, the induced by the connection is preserved; that is

' R(X, Y )Z = R(' X,' Y )' Z, ⇤ ⇤ ⇤ ⇤ as well as the parallel transport, the of a vector field along a curve, the exponential map, sec- tional curvature, and . 746 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS Actually, all concepts that are local in nature are pre- served by local di↵eomorphisms! So, except for the Levi- Civita connection and the Riemann on vectors, all the above concepts are preserved under local di↵eomor- phisms.

Proposition 16.2. If ': M N is a local isometry, then the following concepts are! preserved: (1) The covariant derivative of vector fields along a curve ; that is DX D' X d' = ⇤ , (t) dt dt for any vector field X along , with (' X)(t)= ⇤ d'(t)Y (t), for all t.

(2) Parallel along a curve. If P denotes parallel transport along the curve and if P' denotes parallel transport along the curve ' , then d'(1) P = P' d'(0). 16.1. ISOMETRIES AND LOCAL ISOMETRIES 747 (3) Geodesics. If is a in M, then ' is a geodesic in N. Thus, if v is the unique geodesic with (0) = p and v0 (0) = v, then ' = , v d'pv wherever both sides are defined. Note that the do-

main of d'pv may be strictly larger than the do- main of v. For example, consider the inclusion of an open disc into R2. (4) Exponential maps. We have ' exp =exp d' , p '(p) p wherever both sides are defined. (5) Riemannian curvature tensor. We have

d'pR(x, y)z = R(d'px, d'py)d'pz, for all x, y, z T M. 2 p 748 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS (6) Sectional, Ricci, and Scalar curvature. We have

K(d'px, d'py)=K(x, y)p, for all linearly independent vectors x, y T M; 2 p Ric(d'px, d'py)=Ric(x, y)p for all x, y T M; 2 p S = S '. M N where SM is the scalar curvature on M and SN is the scalar curvature on N.

Ausefulpropertyoflocaldi↵eomorphismsisstatedbe- low. For a proof, see O’Neill [44] (Chapter 3, Proposition 62):

Proposition 16.3. Let ', : M N be two local isometries. If M is connected and! if '(p)= (p) and d' = d for some p M, then ' = . p p 2 16.2. RIEMANNIAN COVERING MAPS 749 16.2 Riemannian Covering Maps

The notion of covering map discussed in Section 7.3 (see Definition 7.8) can be extended to Riemannian manifolds.

Definition 16.1. If M and N are two Riemannian man- ifold, then a map ⇡ : M N is a Riemannian covering i↵the following conditions! hold: (1) The map ⇡ is a smooth covering map. (2) The map ⇡ is a local isometry.

Recall from Section 7.3 that a covering map is a local di↵eomorphism.

AwaytoobtainametriconamanifoldM is to pull-back the g on a N along a local di↵eomor- phism ': M N (see Section 11.2). ! 750 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS If ' is a covering map, then it becomes a Riemannian covering map.

Proposition 16.4. Let ⇡ : M N be a smooth cov- ering map. For any Riemannian! metric g on N, there is a unique metric ⇡⇤g on M, so that ⇡ is a Rieman- nian covering.

In general, if ⇡ : M N is a smooth covering map, a metric on M does not! induce a metric on N such that ⇡ is a Riemannian covering.

However, if N is obtained from M as a quotient by some suitable action (by a group G)onM,thenthe projection ⇡ : M M/G is a Riemannian covering. ! 16.2. RIEMANNIAN COVERING MAPS 751 Because a Riemannian covering map is a local isometry, we have the following useful result.

Proposition 16.5. Let ⇡ : M N be a Riemannian covering. Then, the geodesics! of (M,g) are the pro- jections of the geodesics of (N,h) (curves of the form ⇡ , where is a geodesic in N), and the geodesics of(N,h) are the liftings of the geodesics of (M,h) (curves in N such that ⇡ is a geodesic of (M,h)). As a corollary of Proposition 16.4 and Theorem 7.12, ev- ery connected M has a simply con- nected covering map ⇡ : M M,where⇡ is a Rieman- nian covering. ! f Furthermore, if ⇡ : M N is a Riemannian covering and ': P N is a local! isometry, it is easy to see that its lift ': P! M is also a local isometry. ! e 752 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS In particular, the deck-transformations of a Riemannian covering are isometries.

In general, a local isometry is not a Riemannian cover- ing. However, this is the case when the source space is complete.

Proposition 16.6. Let ⇡ : M N be a local isome- try with N connected. If M is! a complete manifold, then ⇡ is a Riemannian covering map. 16.3. RIEMANNIAN SUBMERSIONS 753 16.3 Riemannian Submersions

Let ⇡ : M B be a surjective submersion between two Riemannian! manifolds (M,g)and(B,h).

1 For every b B,thefibre⇡ (b)isaRiemanniansub- 2 1 manifold of M,andforeveryp ⇡ (b), the tangent 1 1 2 space Tp⇡ (b)to⇡ (b)atp is Ker d⇡p.

The tangent space TpM to M at p splits into the two components

T M =Kerd⇡ (Ker d⇡ )?, p p p where =Kerd⇡ is the vertical subspace of T M Vp p p and =(Kerd⇡ )? (the orthogonal complement of Hp p Vp with respect to the metric gp on TpM)isthehorizontal subspace of TpM. 754 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS Any tangent vector u T M can be written uniquely as 2 p u = u + u , H V with u p and u p. H 2H V 2V

Because ⇡ is a submersion, d⇡p gives a linear between and T B. Hp b

If d⇡p is an isometry, then most of the di↵erential geom- etry of B can be studied by “lifting” from B to M.

Definition 16.2. Amap⇡ : M B between two Rie- mannian manifolds (M,g)and(B,h! )isaRiemannian submersion if the following properties hold: (1) The map ⇡ is surjective and a smooth submersion. 1 (2) For every b B and every p ⇡ (b), the map d⇡p is an isometry2 between the horizontal2 subspace of Hp TpM and TbB. 16.3. RIEMANNIAN SUBMERSIONS 755 We will see later that Riemannian submersions arise when B is a reductive , or when B is ob- tained from a free and proper action of a acting by isometries on B.

Every vector field X on B has a unique horizontal lift 1 X on M,definedsuchthatforeveryp ⇡ (b), 2 1 X(p)=(d⇡p) X(b).

Since d⇡p is an isomorphism between p and TpB,the above condition can be written H

d⇡ X = X ⇡, which means that X and X are ⇡-related (see Definition 6.5).

The following proposition is proved in O’Neill [44] (Chap- ter 7, Lemma 45) and Gallot, Hulin, Lafontaine [23] (Chap- ter 2, Proposition 2.109). 756 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS Proposition 16.7. Let ⇡ : M B be a Rieman- nian submersion between two Riemannian! manifolds (M,g) and (B,h). (1) For any two vector fields X, Y X(B), we have 2 (a) X,Y = X, Y ⇡. h i h i (b) [X,Y ] = [X, Y ]. H (c) ( XY ) = XY , where is the Levi–Civita connectionr H onr M. r

(2) If is a geodesic in M such that 0(0) is a hori- zontal vector, then is horizontal geodesic in M (which means that 0(t) is a horizontal vector for all t), and c = ⇡ is a geodesic in B of the same length than . (3) For every p M, if c is a geodesic in B such that c(0) = ⇡(p),2 then for some ✏ small enough, there is a unique horizonal lift of the restriction of c to [ ✏,✏], and is a geodesic of M. (4) If M is complete, then B is also complete. 16.3. RIEMANNIAN SUBMERSIONS 757 An example of a Riemannian submersion is ⇡ : S2n+1 n n ! CP ,whereS2n+1 has the canonical metric and CP has the Fubini–Study metric.

Remark: It shown in Petersen [45] (Chapter 3, Section 5), that the connection Y on M is given by rX 1 XY = XY + [X,Y ] . r r 2 V 758 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS

16.4 Isometries and Killing Vector Fields ~

Recall from Section ?? that if X is a vector field on a man- q ifold M,thenforany(0,q)-tensor S (M,(T ⇤)⌦ (M)), 2 the LXS of S with respect to X is defined by

d (L S) = (⇤S) , M, X p dt t p 2 t=0 where t is the local one-parameter group associated with X,andthatbyProposition??,

(LXS)(X1,...,Xq)=X(S(X1,...,Xq)) q S(X ,...,[X, X ],...,X ), 1 i q i=1 X for all X ,...,X X(M). 1 q 2 16.4. ISOMETRIES AND KILLING VECTOR FIELDS ~ 759 In particular, if S = g (the ), we get

L g(Y,Z)=X( Y,Z ) [X, Y ],Z Y,[X, Z] , X h i h ih i where we write X, Y and g(X, Y )interchangeably. h i

If t is an isometry (on its domain), then t⇤(g)=g,so LXg =0.

In fact, we have the following result proved in O’Neill [44] (Chapter 9, Proposition 23).

Proposition 16.8. For any vector field X on a Rie- mannian manifold (M,g), the di↵eomorphisms t in- duced by the flow of X are isometries (on their domain) i↵ LXg =0. 760 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS

Informally, Proposition 16.8 says that LXg measures how much the vector field X changes the metric g.

Definition 16.3. Given a Riemannian manifold (M,g), avectorfieldX is a Killing vector field i↵the Lie deriva- tive of the metric vanishes; that is, LXg =0.

Recall from Section ?? (see Proposition ??)thattheco- variant derivative Xg of the Riemannian metric g on a manifold M is givenr by

(g)(Y,Z)=X( Y,Z ) Y,Z Y, Z , rX h i hrX ih rX i for all X, Y, Z X(M), and that the connection on M is compatible2 with g i↵ (g)=0forallX. r rX 16.4. ISOMETRIES AND KILLING VECTOR FIELDS ~ 761 Also, the covariant derivative X of a vector field X is the (1, 1)-tensor defined so thatr ( X)(Y )= X. r rY The above facts imply the following Proposition.

Proposition 16.9. Let (M,g) be a Riemannian man- ifold and let be the Levi–Civita connection on M induced by g.r For every vector field X on M, the following conditions are equivalent:

(1) X is a Killing vector field; that is, LXg =0. (2) X( Y,Z )= [X, Y ],Z + Y,[X, Z] for all Y,Z h i h i h i 2 X(M).

(3) Y X, Z + ZX, Y =0for all Y,Z X(M); thathr is, iX ishr skew-adjointi relative to g.2 r

Condition (3) shows that any parallel vector field is a Killing vector field. 762 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS Remark: It can be shown that if is any geodesic in M,thentherestrictionX of X to is a Jacobi field (see Section 14.5), and that X,0 is constant along (see O’Neill [44], Chapter 9, Lemmah i 26).

Since the Lie derivative LX is R-linear in X and since

[LX,LY ]=L[X,Y ], the Killing vector fields on M form a Lie subalgebra i(M)oftheLiealgebraX(M)ofvectorfieldsonM. K However, unlike X(M), the i(M)isfinite- dimensional. K

In fact, the Lie subalgebra c i(M)ofcompleteKilling K vector fields is anti-isomorphic to the Lie algebra i(M)of the Lie group Isom(M)ofisometriesofM (see Section 11.2 for the definition of Isom(M)). 16.4. ISOMETRIES AND KILLING VECTOR FIELDS ~ 763 The following result is proved in O’Neill [44] (Chapter 9, Lemma 28) and Sakai [49] (Chapter III, Lemma 6.4 and Proposition 6.5).

Proposition 16.10. Let (M,g) be a connected Rie- mannian manifold of dimension n (equip-ped with the Levi–Civita connection on M induced by g). The Lie algebra i(M) of Killing vector fields on M has di- mensionK at most n(n +1)/2.

We also have the following result proved in O’Neill [44] (Chapter 9, Proposition 30) and Sakai [49] (Chapter III, Corollary 6.3).

Proposition 16.11. Let (M,g) be a Riemannian man- ifold of dimension n (equipped with the Levi–Civita connection on M induced by g). If M is complete, then every Killing vector fields on M is complete. 764 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS The relationship between the Lie algebra i(M)andKilling vector fields is obtained as follows.

For every element X in the Lie algebra i(M)ofIsom(M) (viewed as a left- vector field), define the vector field X+ on M by

d X+(p)= (' (p)) ,p M, dt t 2 t=0 where t 't =exp(tX)istheone-parametergroup associated7! with X.

+ Because 't is an isometry of M,thevectorfieldX is a Killing vector field, and it is also easy to show that ('t) is the one-parameter group of X+. 16.4. ISOMETRIES AND KILLING VECTOR FIELDS ~ 765 + Since 't is defined for all t,thevectorfieldX is com- plete. The following result is shown in O’Neill [44] (Chap- ter 9, Proposition 33).

Theorem 16.12. Let (M,g) be a Riemannian mani- fold (equipped with the Levi–Civita connection on M induced by g). The following properties hold: (1) The c i(M) of complete Killing vector fields on M is aK Lie subalgebra of the Lie algebra i(M) of Killing vector fields. K (2) The map X X+ is a Lie anti-isomorphism be- 7! tween i(M) and c i(M), which means that K [X+,Y+]= [X, Y ]+,X,Y i(M). 2

For more on Killing vector fields, see Sakai [49] (Chapter III, Section 6).

In particular, complete Riemannian manifolds for which i(M)hasthemaximumdimensionn(n +1)/2arechar- acterized. 766 CHAPTER 16. ISOMETRIES, SUBMERSIONS, KILLING VECTOR FIELDS