2 tr(H)

To describe the domain of de¬nition of the exponential mapping we consider the

sets

Z h := {x ∈ M : 2

1

trx (H0 ) = 0 and trx (H) < 0},

±

Gh := {x ∈ M : 0 > trx (H0 ) > ’ trx (H)2 and trx (H) < 0}

2

1

±

γ ± (h, h)

= {x ∈ M : ±γ(h, h) 0 for ± 0, trx (H) < 0},

E h := {x ∈ M : ’ trx (H)2 = 2

1

trx (H0 ) and trx (H) < 0}

±

= {x ∈ M : γ ± (h, h) = 0 and trx (H) < 0},

Lh := {x ∈ M : ’ trx (H)2 > 2

1

trx (H0 )}

±

= {x ∈ M : γ ± (h, h) 0 for ± 0},

where γ(h, h) = trx (H 2 ), and γ ± (h, h) = trx (H0 ) + ± trx (H)2 , see (45.5), are the

2

integrands of Gb0 (h, h) and G±0 (h, h), respectively. Then we consider the numbers

b

4

z h := inf ’ : x ∈ Zh ,

trx (H)

2

’± trx (H) ’ ’± trx (H0 )

h

: x ∈ Gh

g := inf 4 ,

2 ) + ± tr(H)2

trx (H0

2

eh := inf ’ : x ∈ Eh ,

trx (H)

2

’± trx (H) ’ ’± trx (H0 )

h

: x ∈ Lh

l := inf 4 ,

2 ) + ± tr(H)2

trx (H0

if the corresponding set is not empty, with value ∞ if the set is empty. Denote

mh := inf{z h , g h , eh , lh }. Then exp±0 (th) is maximally de¬ned for t ∈ [0, mh ).

b

The second representations of the sets Gh , Lh , and E h clari¬es how to take care of

timelike, spacelike, and lightlike vectors, respectively.

Proof. Using X(t) := g ’1 gt the geodesic equation reads as

±n ’ 1

1 1

tr(X 2 ) Id + tr(X)2 Id,

X = ’ tr(X)X + 2

2 4±n 4±n

and it is easy to see that a solution X satis¬es

1

X0 = ’ tr(X)X0 .

2

45.11

45.12 45. Manifolds of Riemannian metrics 493

Then X(t) is in the plane generated by H0 and Id for all t and the solution has the

form g(t) = b0 exp(a(t) Id +b(t)H0 ). Since gt = g(t)(a (t) Id +b (t)H0 ) we have

X(t) = a (t) Id +b (t)H0 and

X (t) = a (t) Id +b (t)H0 ,

and the geodesic equation becomes

1

a (t) Id +b (t)H0 = ’ na (t)(a (t) Id +b (t)H0 )+

2

1

(na (t)2 + b (t)2 tr(H0 )) Id +

2

+

4±n

±n ’ 1 2

(n a (t)2 ) Id .

+ 2

4±n

We may assume that Id and H0 are linearly independent; if not H0 = 0 and b(t) = 0.

Hence, the geodesic equation reduces to the di¬erential equation

2

±

n tr(H0 )

2

(b )2

a = ’ (a ) +

4 4±n

b = ’na b

2

tr(H)

with initial conditions a(0) = b(0) = 0, a (0) = n, and b (0) = 1.

If we take p(t) = exp( n a) it is easy to see that then p should be a solution of p =0

2

and from the initial conditions

t2

t

p(t) = 1 + tr(H) + (tr(H)2 + ±’1 tr(H0 )).

2

2 16

Using that the second equation becomes b = p’1 , and then b is obtained just

by computing the integral. The solutions are de¬ned in [0, mh ) where mh is the

in¬mum over the support of h of the ¬rst positive root of the polynomial p, if it

exists, and ∞ otherwise. The description of mh is now a technical fact.

45.12. The exponential mapping. For b0 ∈ GL(Tx M, Tx M ) and H = (b0 )’1 h

—

—

let Cb0 be the subset of L(Tx M, Tx M ) given by the union of the sets (compare with

Z h , Gh , E h , Lh from (45.11))

2

{h : tr(H0 ) = 0, tr(H) ¤ ’4},

2

’± tr(H) ’ ’± tr(H0 )

2 2

1

> ’ tr(H) , 4 ¤ 1, tr(H) < 0 ,

h:0> tr(H0 ) 2

± tr(H0 ) + ± tr(H)2

h : ’ tr(H)2 = 2

1

tr(H0 ), tr(H) < ’2 ,

±

closure

2

’± tr(H) ’ ’± tr(H0 )

h : ’ tr(H)2 > 2

1

¤1

tr(H0 ), 4 ,

2

± ± tr(H)2

tr(H0 ) +

45.12

494 Chapter IX. Manifolds of mappings 45.12

which by some limit considerations coincides with the union of the following two

sets:

closure

2

’± tr(H) ’ ’± tr(H0 )

tr(H0 ) > ’ tr(H)2 , 4

2

1

¤ 1, tr(H) < 0

h:0> ,

2

± ± tr(H)2

tr(H0 ) +

closure

2

’± tr(H) ’ ’± tr(H0 )

h : ’ tr(H)2 > 2

1

¤1

tr(H0 ), 4 .

2

± ± tr(H)2

tr(H0 ) +

—

So Cb0 is closed. We consider the open sets Ub0 := L(Tx M, Tx M ) \ Cb0 , Ub0 :=

{(b0 )’1 h : h ∈ Ub0 } ‚ L(Tx M, Tx M ), and ¬nally the open sub ¬ber bundles over

GL(T M, T — M )

{b0 } — Ub0 : b0 ∈ GL(T M, T — M ) ‚ GL(T M, T — M ) —M L(T M, T — M ),

U :=

{b0 } — Ub0 : b0 ∈ GL(T M, T — M ) ‚ GL(T M, T — M ) —M L(T M, T M ).

U :=

Then we consider the mapping ¦ : U ’ GL(T M, T — M ) which is given by the

following composition

Id —M exp

•

U ’ U ’ GL(T M, T — M ) —M L(T M, T M ) ’ ’ ’ ’

’ ’ ’’’

Id —M exp

’ ’ ’ ’ GL(T M, T — M ) —M GL(T M, T M ) ’ GL(T M, T — M ),

’’’ ’

where (b0 , h) := (b0 , (b0 )’1 h) is a ¬ber respecting di¬eomorphism, (b0 , H) := b0 H

is a di¬eomorphism for ¬xed b0 , and where the other two mappings will be discussed

below.

The usual ¬berwise exponential mapping

exp : L(T M, T M ) ’ GL(T M, T M )

is a di¬eomorphism near the zero section on the ball of radius π centered at zero

in a norm on the Lie algebra for which the Lie bracket is sub multiplicative, for

example. If we ¬x a symmetric positive de¬nite inner product g, then exp restricts

to a global di¬eomorphism from the linear subspace of g-symmetric endomorphisms

onto the open subset of matrices which are positive de¬nite with respect to g. If g

has signature this is no longer true since then g-symmetric matrices may have non

real eigenvalues.

On the open set of all matrices whose eigenvalues » satisfy | Im »| < π, the expo-

nential mapping is a di¬eomorphism, see [Varadarajan, 1977].

The smooth mapping • : U ’ GL(T M, T — M ) —M L(T M, T M ) is given by

•(b0 , H) := (b0 , a±,H (1) Id +b±,H (1)H0 ) (see theorem (45.11)). It is a di¬eomor-

phism onto its image with the following inverse:

√ ’1

±

2

± tr(H0 )

tr(H)

4 e 4 cos ’ 1 Id +

n 4

√ ’1

2

± tr(H0 )

tr(H)

ψ(H) := 4

+ √ ’1 2

e 4 sin H0 if tr(H0 ) = 0

4

2

± tr(H0 )

tr(H)

4

e 4 ’ 1 Id otherwise,

n

45.12

45.15 45. Manifolds of Riemannian metrics 495

where cos is considered as a complex function, cos(iz) = i cosh(z).

The mapping (pr1 , ¦) : U ’ GL(T M, T — M ) —M GL(T M, T — M ) is a di¬eomor-

phism on an open neighborhood of the zero section in U .

45.13. Theorem. In the setting of (45.12) the exponential mapping exp±0 for the

b

±

metric G is a real analytic mapping de¬ned on the open subset

Ub0 := {h ∈ Cc (L(T M, T — M )) : (b0 , h)(M ) ‚ U },

∞

and it is given by

expb0 (h) = ¦ —¦ (b0 , h).

The mapping (πB , exp) : T B ’ B—B is a real analytic di¬eomorphism from an open

neighborhood of the zero section in T B onto an open neighborhood of the diagonal

in B — B. Ub0 is the maximal domain of de¬nition for the exponential mapping.

Proof. Since B is a disjoint union of chart neighborhoods, it is trivially a real

analytic manifold, even if M is not supposed to carry a real analytic structure.

From the consideration in (45.12) it follows that exp = ¦— and (πM , exp) are just

push forwards by real analytic ¬ber respecting mappings of sections of bundles. So

by (30.10) they are smooth, and this applies also to their inverses.

To show that these mappings are real analytic, by (10.3) it remains to check that

they map real analytic curves into real analytic curves. These are described in

(30.15). It is clear that ¦ has a ¬berwise extension to a holomorphic germ since ¦

is ¬ber respecting from an open subset in a vector bundle and is ¬berwise a real

analytic mapping. So the push forward ¦— preserves the description (30.15) and

maps real analytic curves to real analytic curves.

45.14. Submanifolds of pseudo Riemannian metrics. We denote by Mq

the space of all pseudo Riemannian metrics on the manifold M of signature (the

dimension of a maximal negative de¬nite subspace) q. It is an open set in a closed

locally a¬ne subspace of B and thus a splitting submanifold of it with tangent

bundle T Mq = Mq — Cc (M ← S 2 T — M ).

∞

We consider a geodesic c(t) = c0 e(a(t) Id +b(t)H0 ) for the metric G± in B starting

at c0 in the direction of h as in (45.11). If c0 ∈ Mq then h ∈ Tc0 Mq if and

only if H = (c0 )’1 h ∈ Lsym,c0 (T M, T M ) is symmetric with respect to the pseudo

Riemannian metric c0 . But then e(a(t) Id +b(t)H0 ) ∈ Lsym,c0 (T M, T M ) for all t in

the domain of de¬nition of the geodesic, so c(t) is a curve of pseudo Riemannian

metrics and thus of the same signature q as c0 . Thus, we have

45.15. Theorem. For each q ¤ n = dim M the submanifold Mq of pseudo

Riemannian metrics of signature q on M is a geodesically closed submanifold of

(B, G± ) for each ± = 0.

45.15

496 Chapter IX. Manifolds of mappings 45.17

1

Remark. The geodesics of (M0 , G± ) have been studied for ± = n , in [Freed,

Groisser, 1989], [Gil-Medrano, Michor, 1991] and from (45.15) and (45.11) we see

that they are completely analogous for every positive ±.

For ¬xed x ∈ M there exists a family of homothetic pseudo metrics on the ¬nite

2—

dimensional manifold S+ Tx M whose geodesics are given by the evaluation of the

geodesics of (M0 , G± ) (see [Gil-Medrano, Michor, 1991] for more details). When

± is negative, it is not di¬cult to see, from (45.15) and (45.11) again, that a

geodesic of (M0 , G± ) is de¬ned for all t if and only if the initial velocity h satis¬es

γ ± (h, h) ¤ 0 and tr H > 0 at each point of M , and then the same is true for all

2—

the pseudo metrics on S+ Tx M. These results appear already in [DeWitt, 1967] for

n = 3.

45.16. The local signature of G± . Since G± operates in in¬nite dimensional

spaces, the usual de¬nition of signature is not applicable. But for ¬xed g ∈ Mq the

signature of

’1 ’1 ’1 ’1

±

γgx (hx , kx ) = tr((gx hx )0 (gx kx )0 ) + ± tr(gx kx ) tr(gx kx )

2— —

on Tg (Sq Tx M ) = S 2 Tx M is independent of x ∈ M and the special choice of

g ∈ Mq . We will call it the local signature of G± .

45.17. Lemma. The signature of the quadratic form of (45.16) is

0 for ± > 0

Q(±, q) = q(q ’ n) +

1 for ± < 0.

This result is due to [Schmidt, 1989].

Proof. Since the signature is constant on connected components we have to de-

1 1

termine it only for ± = n and ± = n ’ 1. In a basis for T M and its dual basis

for T — M the bilinear form h ∈ S 2 Tx M has a symmetric matrix. If the basis is

—

orthonormal for g we have (for At = A and C t = C)

’ Idq ’A ’B

0 A B

H = g ’1 h = = ,

Bt Bt

0 Idn’q C C

which describes a typical matrix in the space Lsym,g (Tx M, Tx M ) of all matrices

H ∈ L(Tx M, Tx M ) which are symmetric with respect to gx .

1

Now we treat the case ± = n . The standard inner product tr(HK t ) is positive

de¬nite on Lsym,g (Tx M, Tx M ), and the linear mapping K ’ K t has an eigenspace

of dimension q(n ’ q) for the eigenvalue ’1 in it and a complementary eigenspace

for the eigenvalue 1. So tr(HK) has signature q(n ’ q).

1

For the case ± = n ’1 we again split the space Lsym,g (Tx M, Tx M ) into the subspace

with 0 on the main diagonal, where γg (h, k) = tr(HK) and where K ’ K t has

±

again an eigenspace of dimension q(n ’ q) for the eigenvalue ’1, and the space

±

of diagonal matrices. There γg has signature 1 as determined in the proof of

(45.5).

45.17

46 46. The Korteweg “ De Vries equation as a geodesic equation 497

45.18. The submanifold of almost symplectic structures.

A 2-form ω ∈ „¦2 (M ) = C ∞ (M ← Λ2 T — M ) can be non degenerate only if M is

of even dimension dim M = n = 2m. Then ω is non degenerate if and only if

ω § · · · § ω = ω m is nowhere vanishing. Usually this latter 2m-form is regarded as

the volume form associated with ω, but a short computation shows that we have

m

1

m! |ω |.

vol(ω) =

tr(ω ’1 •)ω m for all • ∈ „¦2 (M ).

1

This implies m• § ω m’1 = 2

45.19. Theorem. The space „¦2 (M ) of non degenerate 2-forms is a splitting

nd

geodesically closed submanifold of (B, G± ) for each ± = 0.

Proof. We consider a geodesic c(t) = c0 e(a(t) Id +b(t)H0 ) for the metric G± in B

starting at c0 in the direction of h as in (45.11). If c0 = ω ∈ „¦2 (M ) then h ∈

nd

’1

2

„¦c (M ) if and only if H = ω h is symmetric with respect to ω, since we have

ω(HX, Y ) = ωω ’1 hX, Y = hX, Y = h(X, Y ) = ’h(Y, X) = ’ω(HY, X) =

ω(X, HY ). At a point x ∈ M we may choose a Darboux frame (ei ) such that

ω(X, Y ) = Y t JX where

0 Id

J= .

’ Id 0