Quarterly Journal of Mathematics (Oxford) 42 (1991), 183{202
THE RIEMANNIAN MANIFOLD
OF ALL RIEMANNIAN METRICS
Olga Gil-Medrano, Peter W. Michor
Introduction
If M is a (not necessarily compact) smooth nite dimensional manifold,
the space M = M(M) of all Riemannian metrics on it can be endowed
with a structure of an in nite dimensional smooth manifold modeled on the
;
0
space D(S2T M) of symmetric -tensor elds with compact support, in the
2
1 2
sense of [Michor, 1980]. The tangent bundle of M is TM = C (S+T M)
D(S2T M) and a smooth Riemannian metric can be de ned by
Z
;1 ; 1
Gg(h k) = tr(g hg k) vol(g):
M
In this paper we study the geometry of (M G) by using the ideas developed
in [Michor, 1980].
With that di erentiable structure on M it is possible to use variational
principles and so we start in section 2 by computing geodesics as the curves
in M minimizing the energy functional. From the geodesic equation, the
covariant derivative of the Levi-Civita connection can be obtained, and that
provides a direct method for computing the curvature of the manifold.
Christo el symbol and curvature turn out to be pointwise in M and so,
although the mappings involved in the de nition of the Ricci tensor and the
scalar curvature have no trace, in our case we can de ne the concepts of "Ricci
like curvature" and "scalar like curvature".
The pointwise character mentioned above allows us in section 3, to solve
explicitly the geodesic equation and to obtain the domain of de nition of the
The rst author was partially supported the CICYT grant n. PS87-0115-G03-01. This
paper was prepared during a stay of the second author in Valencia, by a grant given by
Conseller a de Cultura, Educaci on y Ciencia, Generalidad Valenciana.
Typeset by AMS-TEX
Typeset by AMS-TEX
2 Olga Gil-Medrano, Peter W. Michor
exponential mapping. That domain turns out to be open for the topology
considered on M and the exponential mapping is a di eomorphism onto its
image which is also explicitly given. In the L2-topology given by G itself this
domain is, however, nowhere open. Moreover, we prove that it is, in fact, a
real analytic di eomorphism, using [Kriegl-Michor, 1990]. We think that this
exponential mapping will be a very powerful tool for further investigations of
the strati cation of orbit space of M under the di eomorphism group, and
also the strati cation of principal connections modulo the gauge group.
In section 4 Jacobi elds of an in nite dimensional Riemannian manifold
are de ned as the in nitesimal geodesic variations and we show that they
must satisfy the Jacobi Equation. For the manifold (M G) the existence of
Jacobi elds, with any initial conditions, is obtained from the results about
the exponential mapping in section 3. Uniqueness and the fact that they
are exactly the solutions of the Jacobi Equation follows from its pointwise
character. We nally give the expresion of the Jacobi elds.
For xed x 2 M, there exists a family of homothetic Riemannian metrics
2
in the nite dimensional manifold S+ Tx M whose geodesics are given by the
evaluation of the geodesics of (M G). The relationship between the geometry
of (M G) and that of these manifolds is explained in each case and it is used
to visualize the exponential mapping. Nevertheless, in this paper, we have not
made use of these manifolds to obtain the results, every computation having
been made directly on the in nite dimensional manifold.
2
Metrics on S+Tx M for three dimensional manifolds M which are similar
to ours but have di erent signatures were considered by [DeWitt, 1967]. He
computed the curvature and the geodesics and gave some ideas on howto use
them to determine the distance between two 3-geometries, but without con-
sidering explicitly the in nite dimensional manifold of all Riemannian metrics
on a given manifold.
The topology of M(M), under the assumption that M is compact, ori-
entable, without boundary, was studied by [Ebin, 1970] who treated G in the
context of Sobolev completions of mapping spaces and computed the Levi-
Civita connection. In the same context and under the same assumptions, the
curvature and the geodesics have been computed in [Freed-Groisser, 1989].
The explicit formulas of the three papers just mentioned are the same as
in this paper.
We want to thank A. Montesinos Amilibia for producing the computer
image of gure 1.
The Riemannian manifold of all Riemannian metrics 3
1. The general setup
1.1. The space of Riemannian metrics. Let M be a smooth second
countable nite dimensional manifold. Let S2T M denote the vector bundle
;
0
2
of all symmetric -tensors on M and let S+T M be the open subset of all
2
the positive de nite ones. Then the space M(M) = M of all Riemannian
1 2
metrics is the space of sections C (S+T M) of this ber bundle. It is open
1 1
in the space of sections C (S2T M) in the Whitney C -topology, in which
1
the latter space is, however, not a topological vector space, since h converges
n
to 0 if and only if h has compact support. So the space D = D(S2T M) of
sections with compact support is the largest topological vector space contained
1 1
in the topological group (C (S2T M) +), and the trace of the Whitney C -
topology on it coincides with the inductive limit topology
1
D(S2T M) = lim CK (S2T M)
;
!
K
1
where CK (S2T M) is the space of all sections with support contained in K
and where K runs through all compact subsets of M.
1 2
So we declare the path components of C (M S+T M) for the Whitney
1
C -topology also to be open. We get a topology which is ner than the
Whitney topology, where each connected component is homeomorphic to an
1 2
open subset in D = D(S2T M). So M = C (S+T M) is a smooth manifold
modeled on nuclear (LF)-spaces, and the tangent bundle is given by TM =
M D.
1.2. Remarks. The main reference for the in nite dimensional manifold
structures is [Michor, 1980]. But the di erential calculus used there is not
completely up to date, the reader should consult [Fr
olicher-Kriegl, 1988],
whose calculus is more natural and much easier to apply. There a mapping
between locally convex spaces is smooth if and only if it maps smooth curves
to smooth curves. See also [Kriegl-Michor, 1990] for a setting for real analytic
mappings along the same lines and applications to manifolds of mappings.
As a nal remark let us add that the di erential structure on the space M
of Riemannian metrics is not completely satisfying, if M is not compact. In
1 1
fact C (S2T M) is a topological vector space with the compact C -topology,
1 2
but the space M = C (S+T M) of Riemannian metrics is not open in it.
Nevertheless, we will see later that the exponential mapping for the natural
Riemannian metric on M is de ned also for some tangent vectors which are
not in D. This is an indication that the most natural setting for manifolds of
1
mappings is based on the compact C -topology, but that one loses existence
4 Olga Gil-Medrano, Peter W. Michor
of charts. In [Michor, 1985] a setting for in nite dimensional manifolds is
presented which is based on an axiomatic structure of smooth curves instead
of charts.
1 2
1.3. The metric. The tangent bundle of the space M = C (S+T M) of
1 2
Riemannian metrics is TM = M D = C (S+T M) D(S2T M). We
identify the vector bundle S2T M with the subbundle
f` 2 L(TM T M) : `t = `g
of L(TM T M), where the transposed is given by the composition
i `
`t : TM ; T M ; T M:
! !
Then the berwise inner product on S2T M induced by g 2 M is given by
;1 ;1
the expression hh kig := tr(g hg k), so a smooth Riemannian metric on
M is given by
Z
; 1 ; 1
Gg(h k) = tr(g hg k) vol(g)
M
where vol(g) is the positive density de ned by the local formula vol(g) =
p
det gdx. We call this the canonical Riemannian metric on M, since it is
invariant under the action of the di eomorphism group Di (M) on the space
M of metrics. The integral is de ned since h, k have compact support. The
metric is positive de nite, Gg(h h) 0 and Gg(h h) = 0 only if h =0. So Gg
de nes a linear injective mapping from the tangent space TgM = D(S2T M)
0
into its dual D(S2T M) , the space of distributional densities with values in
the dual bundle S2TM. This linear mapping is, however, never surjective, so
G is only a weak Riemannian metric. The tangent space TgM = D(S2T M) is
a pre-Hilbert space, whose completion is a Sobolev space of order 0, depending
on g if M is not compact.
1.4. Remark. Since G is only a weak Riemannian metric, all objects which
are only implicitly given, a priori lie in the Sobolev completions of the relevant
spaces. In particular this applies to the formula
2G( r ) = G( ) + G( ) ; G( )
+ G([ ] ) + G([ ] ) ; G([ ] )
which a priori gives only uniqueness but not existence of the Levi Civita
covariant derivative.
The Riemannian manifold of all Riemannian metrics 5
2. Geodesics, Levi Civita connection,
and curvature
2.1 The covariant derivative. Since we will need later the covariant deriv-
ative of vector elds along a geodesic for the derivation of the Jacobi equation,
we present here a careful description of the notion of the covariant derivative,
which is valid in in nite dimensions. Here M might be any in nite dimen-
sional manifold, modeled on locally convex spaces. If we are given a horizontal
bundle, complementary to the vertical one, in T2M, with the usual proper-
ties of a linear connection, then the projection from T2M to the vertical
bundle V(TM) along the horizontal bundle, followed by the vertical projec-
tion V(TM) ! TM, de nes the connector K : T2M! TM, which has the
following properties:
(1) It is a left inverse to the vertical lift mapping with any foot point.
(2) It is linear for both vector bundle structures on T2M.
(3) The connection is symmetric (torsionfree) if and only if K = K,
where is the canonical ip mapping on the second tangent bundle.
If a connector K is given, the covariant derivative is de ned as follows: Let
f : N !M be a smooth mapping, let s : N ! TM be a vector eld along f
and let Xx 2 TxN. Then
rXx s := (K Ts)(Xx):
In a chart the Christo el symbol is related to the connector by
K(g h k `) = (g ` ; ;g(h k))
.
We want to state one property, which is usually stated rather clumsily in
the literature: If f1 : P !N is another smooth mapping and Yy 2 TyP, then
we have rYy (s f1 ) = rTy ( f1 )Yy s. Equivalently, if vector elds Y 2 X(P) and
X 2 X(N) are f1 -related, then rY (s f1 ) = (rX s) f1 .
@
If V(t) is a vector eld along a smooth curve g(t), we have r@t V = V ;
@t
;g(gt V), in local coordinates.
If c : R2 !M is a smooth mapping for a symmetric connector K we have
@
r@t c(t s) = K T(Tc @s) @t = K T2c T(@s) @t
@s
= K T2c T(@s) @t = K T2c T(@s) @t
@
= K T2c T(@t) @s = r@s c(t s)
@t
which will be used for Jacobi elds.
6 Olga Gil-Medrano, Peter W. Michor
2
2.2. Let t ! g(t) be a smooth curve in M: so g : R M ! S+T M is
7
smooth and by the choice of the topology on M made in 1.1 the curve g(t)
varies only in a compact subset of M, locally in t, by [Michor, 1980, 4.4.4,
4.11, and 11.9]. Then its energy is given by
Z
b
b 1
Ea(g) : = Gg(gt gt)dt
2
a
Z Z
b
1 ; 1 ;1
= tr(g gtg gt) vol(g) dt
2
a M
@
where gt = g(t).
@t
Nowwe consider a variation of this curve, so we assume now that (t s) !
7
g(t s) is smooth in all variables and locally in (t s) it only varies within a
compact subset in M | this is again the e ect of the topology chosen in 1.1.
Note that g(t 0) is the old g(t) above.
2.3. Lemma. In the setting of 2. 2 we have the rst variation formula
@
b t=b
j Ea(g( s)) = Gg(gt gs)j +
0
t=a
@s
Z
b
;
;1 1 ;1 ;1 1 ;1
+ Gg ;gtt + gtg gt + tr(g gtg gt)g ; tr(g gt)gt gs dt:
4 2
a
Proof. We have
Z Z
b
@ @ 1
b ;1 ;1
j Ea(g( s)) = j tr(g gtg gt) vol(g)dt:
0 0
@s @s 2
a M
@
We may interchange j with the rst integral since this is nite dimensional
0
@s
R
analysis, and we may interchange it with the second one, since is a con-
M
tinuous linear functional on the space of all smooth densities with compact
support on M, by the chain rule. Then we use that tr is linear and contin-
1 ; 1 ;1 ;1 ;1
uous, d(vol)(g)h = tr(g h) vol(g), and that d(( ) ) (g)h = ;g hg
2
and partial integration.
2.4 The geodesic equation. By lemma 2.3 the curve t ! g(t) is a geodesic
7
if and only if we have
;1 1 ;1 ;1 1 ;1
gtt = gtg gt + tr(g gtg gt) g ; tr(g gt) gt
4 2
=;g(gt gt)
The Riemannian manifold of all Riemannian metrics 7
where the Christo el symbol ; : M D D ! D is given by symmetrisation
1 ;1 1 ;1
;g(h k) = hg k + kg h+
2 2
1 ;1 ;1 1 ;1 1 ;1
+ tr(g hg k) g ; tr(g h) k ; tr(g k) h:
4 4 4
The sign of ; is chosen in such a way that the horizontal subspace of T2M is
parameterized by (x y z ;x(y z)). If instead of the obvious framing we use
;1
TM = M D 3 (g h) ! (g g h) =: (g H) 2 fgg D(Lsym g(TM TM))
7
M D(L(TM TM)), the Christo el symbol looks like
1 1 1 1
;g(H K) = (HK + KH) + tr(HK)Id ; tr(H)K ; tr(K)H
2 4 4 4
;1
and the geodesic equation for H(t) := g gt becomes
@
; 1 1 1
Ht = j (g gt) = tr(HH)Id ; tr(H)H:
0
4 2
@t
2.5 The curvature. In the setting of 2.1, for vector elds X, Y 2 X(N) and
a vector eld s : N ! TM along f : N !M we have
R(X Y)s =(r[ X Y ] ; [rX rY ])s
=(K TK ; K TK ) T2s TX Y
TM
which in local coordinates reduces to the usual formula
R(h k)` = d;(h)(k `) ; d;(k)(h `) ; ;(h ;(k `)) + ;(k ;(h `)):
A global derivation of this formula can be found in [Kainz-Michor, 1987]
2.6 Proposition. The Riemannian curvature for the canonical Riemannian
metric on the manifold M of all Riemannian metrics is given by
;1 1
g Rg(h k)` = [[H K] L]
4
dim M
+ (tr(KL)H ; tr(HL)K)
16
1
+ (tr(H) tr(L)K ; tr(K) tr(L)H)
16
1
+ (tr(K) tr(HL) ; tr(H) tr(KL))Id:
16
Proof. This is a long but elementary computation using the formula from2.5
and
1 ;1 ;1 1 ;1 ;1 1 ;1 ; 1 ;1
d;(h)(k `) = ; kg hg ` ; `g hg k ; tr(g hg kg `)g
2 2 4
1 ;1 ;1 ;1 1 ; 1 ;1
; tr(g kg hg `)g + tr(g kg `)h
4 4
1 ;1 ;1 1 ;1 ;1
+ tr(g hg k)` + tr(g hg `)k:
4 4
8 Olga Gil-Medrano, Peter W. Michor
2.7. Ricci curvature for the Riemannian space (M G) does not exist,
since the mapping k ! Rg(h k)` is just the push forward of the section by
7
a certain tensor eld, a di erential operator of order 0. If this is not zero, it
induces a topological linear isomorphism between certain in nite dimensional
subspaces of TgM, and is therefore never of trace class.
2.8. Ricci like curvature. But we may consider the pointwise trace of the
tensorial operator k ! Rg(h k)` which we call the Ricci like curvature and
7
denote by
Ricg(h `)(x) := tr(kx ! Rg(hx kx)`x):
7
Proposition. The Ricci like curvature of (M G) is given by
4 + n(n +1)
Ricg(h `) = (tr(H) tr(L) ; n tr(HL))
32
n
= ; (4 + n(n + 1))hh0 `ig
32
1
where h0 := h ; tr(H)g.
n
Proof. We compute the pointwise trace tr(kx ! Rg(hx kx)`x) and use the
7
following
Lemma. For H, K, and L 2 Lsym(Rn Rn ) we have
tr(K ! [[H K] L]) = tr(H) tr(L) ; n tr(HL):
7
;
0
We can de ne a -tensor eld Ric on M by
2
Z
Ric( )(g) := Ricg( (g) (g)) vol(g)
M
for , 2 X(M), and g 2 M. Then by the proposition we have
n
Ric( ) = ; (4 + n(n + 1))G( )
0
32
1 ;1
where is the vector eld given by (g) := (g) ; tr(g (g))g.
0 0
n
;
1
2.9. Scalar like curvature. By 2.8 there is a unique -tensor eld Ric
1
on M such that G(Ric( ) ) = Ric( ) for all vector elds , 2 X(M),
n
which is given by Ric( ) = ; (4 + n(n + 1)) . Again, for every g 2 M
0
32
the corresponding endomorphism of TgM is a di erential operator of order 0
The Riemannian manifold of all Riemannian metrics 9
and is never of trace class. But we may again form its pointwise trace as a
linear vector bundle endomorphism on S2T M ! M, which is a function on
M. We call it the scalar like curvature of (M G) and denote it by Scalg. It
turns out to be the constant c(n) depending only on the dimension n of M,
because the endomorphism involved is just the projection onto a hyperplane.
We have
n n(n +1)
c(n) = ; (4 + n(n + 1))( ; 1):
32 2
Remark. For xed g 2 M and x 2 M the expression
~
p
~
;1
Gx g(hx kx) := hh kig(x) det(g g)(x)
~
2
gives a Riemannian metric on S+Tx M. It is not di cult to see that the
Ricci like curvature of (M G) at x is just the Ricci curvature of the family
2
of homothetic metrics on S+Tx M obtained by varying g, and that the scalar
~
p
~ 2
;1
curvature of Gx g equals the function c(n)= det(g g)(x) on S+Tx M.
~
3. Analysis of the exponential mapping
3.1. The geodesic equation 2.4 is an ordinary di erential equation and the
2
evolution of g(t)(x) depends only on g(0)(x) and gt(0)(x) and stays in S+Tx M
for each x 2 M.
The geodesic equation can be solved explicitly and we have
3.2. Theorem. Let g0 2 M and h 2 Tg0 M = D. Then the geodesic in M
starting at g0 in the direction of h is the curve
g(t) = g0e( a( t)Id+b( t)H0 )
tr( H)
;1
where H0 is the traceless part of H := (g0) h (i.e. H0 = H ; Id) and
n
1
where a(t) and b(t) 2 C (M) are de ned as follows:
;
2 t n 2
a(t) = log (1 + tr(H))2 + tr(H0 )t2
n 4 16
!
8 p
2
> 4 n tr(H0 ) t
> 2
>
p arctg where tr(H0 ) =0
6
<
2
4 + t tr(H)
n tr(H0 )
b(t) =
>
t
>
2
>
: where tr(H0 ) = 0:
t
1 + tr(H)
4
10 Olga Gil-Medrano, Peter W. Michor
Here arctg is taken to have values in (; ) for the points of the manifold
2 2
where tr(H) 0, and on a point where tr(H) < 0 we de ne
8
4
>
arctg in [0 ) for t 2 [0 ; )
>
>
> 2 tr(H)
! >
p
>
<
2
4
n tr(H0 ) t
for t = ;
arctg =
4 + t tr(H) > 2 tr(H)
>
>
>
>
4
>
:
arctg in ( ) for t 2 (; 1):
2 tr(H)
Let Nh := fx 2 M : H0(x) = 0g, and if Nh = let th := infftr(H)(x) : x 2
6
Nhg. Then the geodesic g(t) is de ned for t 2 [0 1) if Nh = or if th 0,
4
and it is only de ned for t 2 [0 ; ) if th < 0.
th
Proof. Check that g(t) is a solution of the pointwise geodesic equation. Com-
putations leading to this solution can be found in [Freed-Groisser, 1989].
2
3.3. The exponential mapping. For g0 2 S+Tx M we consider the sets
4
Ug0 := S2Tx M n (;1 ; ] g0
n
Lsym g0(TxM TxM) := f` 2 L(TxM TxM) : g0(`X Y) = g0(X `Y)g
L+ (TxM TxM) := f` 2 Lsym g0(TxM TxM) : ` is positive g
sym g0
4
0
Ug := Lsym g0 (TxM TxM) n (;1 ; ] IdTx M
0
n
2
and the ber bundles over S+T M
[
2
0
U := fg0g Ug : g0 2 S+T M
[
2
Lsym := fg0g Lsym g0 (TxM TxM) : g0 2 S+ T M
n o
[
2
L+ := fg0g L+ (TxM TxM) : g0 2 S+T M
sym
sym g0
[
0 0 2
U := fg0g Ug : g0 2 S+T M :
0
2
Then we consider the mapping : U ! S+ T M which is given by the
following composition
] ' exp
[
0 2
U ; U ; Lsym ; ! L+ ; S+ T M
! ! ; !
sym
The Riemannian manifold of all Riemannian metrics 11
;1
where ](g0 h) := (g0 (g0) h) is a ber respecting di eomorphism, where
'(g0 H) := (g0 a(1)Id + b(1)H0) comes from theorem 3.2, where the usual
exponential mapping exp : Lsym g0 ! L+ is a di eomorphism (see for in-
sym g0
stance [Greub-Halperin-Vanstone, 1972, page 26]) with inverse log, and where
[(g0 H) := g0H, a di eomorphism for xed g0.
2 2
We now consider the mapping (pr1 ) : U ! S+T M S+T M. From
M
the expression of b(1) it is easily seen that the image of (pr1 ) is contained
in the following set:
1
2
V : = (g0 g0 exp H) : tr(H0 ) < (4 )2
n
( ! )
2
;1
tr(log((g0) g)) (4 )2
;1
= (g0 g) : tr log((g0) g) ; Id <
n n
Then (pr1 ) : U ! V is a di eomorphism, since the mapping (g0 A) !
0 1
(g0 (A)) is an inverse to ' : U ! (g0 A) : tr(A2) < (4 )2 , where is
0
n
given by:
! !
8 p
tr( A)
> 4 n tr(A2)
0
>
4
> e cos ; 1 Id
>
>
n 4
>
>
>
!
<
p
tr( A)
4 n tr(A2)
(A) =
0
4
+ p e sin A0 if A0 =0
6
>
>
4
>
n tr(A2)
>
0
>
>
>
> tr( A)
4
:
4
e ; 1 Id otherwise.
n
3.4. Theorem. In the setting of 3. 3 the exponential mapping Expg is a real
0
analytic di eomorphism between the open subsets
0
Ug := fh 2 D(S2T M) : (g0 h)(M) Ug
1 2
0
Vg := fg 2 C (S+T M) : (g0 g)(M) V g ; g0 2 D(S2T M)g
and it is given by
Expg (h) = (g0 h):
0
The mapping ( Exp) : TM! M M is a real analytic di eomorphism
M
from the open neighborhood of the zero section
1 2
U := f(g0 h) 2 C (S+T M) D(S2T M) : (g0 h)(M) Ug
12 Olga Gil-Medrano, Peter W. Michor
onto the open neighborhood of the diagonal
1 2 1 2
V := f(g0 g) 2 C (S+T M) C (S+T M) : (g0 g)(M) V
g ; g0 has compact support g:
All these sets are maximal domains of de nition for the exponential mapping
and its inverse.
Proof. Since M 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 3.3 it follows that Exp = and ( Exp)
M
are just push forwards by smooth ber respecting mappings of sections of
bundles. So by [Michor, 1980, 8.7] they are smooth and this applies also to
their inverses.
To show that these mappings are real analytic, by [Kriegl-Michor, 1990] we
have to check that they map real analytic curves into real analytic curves. So
we may just invoke the description [Kriegl-Michor, 1990, 7.7.2] of real analytic
curves in spaces of smooth sections:
For a smooth vector bundle (E p M) a curve c : R ! C1 (E) is real
analytic if and only if c : R M ! E satis es the following condition:
^
(1) For each n there is an open neighborhood Un of R M in C M and a
(unique) Cn-extension c : Un ! EC such that c( x) is holomorphic
~ ~
for all x 2 M.
In statement (1) the space of sections C1 (E) is equipped with the compact
1
C1 -topology. So we have to show that (1) remains true for the space Cc (E)
of smooth sections with compact support with its inductive limit topology.
1
This is easily seen since we may rst exchange C (E) by the closed linear
1
subspace CK (E) of sections with support in a xed compact subset then we
just note that (1) is invariant under passing to the strict inductive limit in
question.
Now 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 maps real analytic curves
to real analytic curves.
3.5. Remarks. The domain Ug0 of de nition of the exponential mapping
0
does not contain any ball centered at 0 for the norm derived from Gg .
Note that Expg is in fact de ned on the set
0
0 1
Ug := fh 2 C (S2T M) : (g0 h)(M) Ug
0
The Riemannian manifold of all Riemannian metrics 13
which is not contained in the tangent space for the di erentiable structure
we use. Recall now the remarks from 1.2. If we equip M with the compact
1 1
C -topology, then M it is not open in C (S2T M). The tangent space
is then the set of all tangent vectors to curves in M which are smooth in
1
C (S2T M), which is probably not a vector space. The integral in the
de nition of the canonical Riemannian metric might not converge on all these
tangent vectors.
The approach presented here is clean and conceptually clear, but some of
the concepts have larger domains of de nition.
3.6. Visualizing the exponential mapping. Let us x a point x 2 M and
2
let us consider the space S+Tx M = L+(TxM Tx M) of all positive de nite
2
symmetric inner products on TxM. If we x an element g 2 S+Tx M, we may
~
2
de ne a Riemannian metric G on S+Tx M by
p
;1 ;1
;1
Gg(h k) := tr(g hg k) det(g g)
~
2 2
for g 2 S+Tx M and h, k 2 Tg(S+Tx M) = S2Tx M. The variational method
used in section 2 leading to the geodesic equation shows that the geodesic
starting at g0 in the direction h 2 S2Tx M is given by
g(t) = g0e( a( t)Id+b( t)H0 )
in the setting of theorem 3.2. We have the following di eomorphisms
exp [=( g0) 2
Lsym g0 (TxM TxM) ; ! L+ (TxM TxM) ; ! S+Tx M
; ;;;;
sym g0
H ! exp(H) = eH ! g0eH
7 7
2
the manifolds (S+Tx M G) and (Lsym g0 (TxM TxM) ((g0) exp) G) are
isometric. For L 2 Lsym g0 (TxM TxM) we have by the a ne structure
TLLsym g0 (TxM TxM) = Lsym g0 (TxM TxM) and we get
p
;1
((g0) exp) G)L(H K) = det(g g0)hH AL(K)i
~
where hH Ki =tr(HK) and
1
X
1 2(ad(L))2k
tr( L)
2
AL = e :
(2k + 2)!
k=0
14 Olga Gil-Medrano, Peter W. Michor
Figure 1
The geodesic on Lsym g0 (TxM TxM) for that metric starting at 0 in the di-
rection H is then given by
L(t) := a(t)Id + b(t)H0:
2
n
Let us choose now a g0-orthonormal basis of TxM and let g = n g0.
~
2
Then the exponential mapping at g0 of (S+Tx M G) can be viewed as the
0
exponential mapping at 0 of (Matsym(n R) h i ), where for symmetric
The Riemannian manifold of all Riemannian metrics 15
matrices A, B, and C we have
1
0
hA BiC = hA AC (B)i
n
0
where the scale factor is chosen in such a way that we have hId Idi0 = 1.
This exponential mapping is de ned on the set
00
U := fA 2 Matsym(n R) : A0 =0 or A = Id with > ;4g
6
by the formula
;
2 1 n
exp0(A) = log (1 + tr(A))2 + tr(A2) Id
0
n 4 16
!
p
4 n tr(A2)
0
+ p arctg A0:
4 +tr(A)
n tr(A2)
0
If A is traceless (i.e. A0 = A) and if P is the plane in Matsym(n R) through
00
0, Id, and A, then Exp0(P \ U ) P and we can view at a 2-dimensional
picture of this exponential mapping. If we normalize A in such a way that
tr(A2) = n the exponential mapping is just the di eomorphism
0
4 4
4
R2 n (f0g (;1 ; ]) ! (; ) R
n
n n
8
nx
>
4
<
u(x y) = arctg
n
4 + ny
; ;
>
: 2 1
v(x y) = log (4 + ny)2 + n2x2 :
n 16
Here arctg is taken to have values in (0 ) for x 0 and to have values in
(; 0) for x 0. The images of the straight lines and the circles can be seen
in gure 1. They correspond respectively to the images of the geodesics and
;1 4
to the level sets of the distance function dist(0 ) (r). For r < they are
n
exactly the geodesic spheres.
It is known that for a nite dimensional Riemannian manifold, if for some
point the exponential map is de ned in the whole tangent space, then every
other point can be joined with that by a minimizing geodesic (Hopf-Rinow-
theorem). Here we have a nice example where, although only a half line lacks
from the domain of de nition of the exponential mapping, it is far from being
surjective.
16 Olga Gil-Medrano, Peter W. Michor
4. Jacobi elds
4.1. The concept of Jacobi elds. Let (M G) be an in nite dimensional
Riemannian manifold which admits a smooth Levi Civita connection, and let
c : [0 a] !M be a geodesic segment. By a geodesic variation of c we mean a
smooth mapping : [0 a] (;" ") !M such that for each xed s 2 (;" ")
the curve t ! (t s) is a geodesic and (t 0) = c(t).
7
De nition. Avector eld along a geodesic segment c : [0 a] !M is called
a Jacobi eld if and only if it is an in nitesimal geodesic variation of c, i.e.
if there exists a geodesic variation : [0 a] (;" ") ! M of c such that
@
(t) = j (t s).
0
@s
In 2.1 and 2.5 we have set up all the machinery necessary for the usual
proof that any Jacobi eld along a geodesic c satis es the Jacobi equation
0
r@t r@t = R( c )c0:
In a nite dimensional manifold solutions of the Jacobi equation are Jacobi
elds, but in in nite dimensions one has in general neither existence nor
uniqueness of ordinary di erential equations, nor an exponential mapping.
Nevertheless for the manifold (M(M) G) we will show existence and also
uniqueness of solutions of the Jacobi equation for given initial conditions, and
that each solution is a Jacobi eld.
4.2. Lemma. Let g(t) be a geodesic in M(M). Then for a vector eld
along g the Jacobi equation has the following form:
1
= ;gtg;1 g; 1 gt + gtg; 1 + g;1 gt + tr(g;1 gtg;1 ) g
tt t t t
2
1 1 1
; tr(g;1 gtg;1 gtg; 1 ) g + tr(g;1 gtg;1 ) gt ; tr(g; 1 ) gt
t
2 2 2
1 1
+ tr(g;1 gtg;1 gt) ; tr(g;1 gt) :
t
4 2
Proof. From 2.1 we have r@t = ; ;g(gt ), thus
t
r@t r@t = ; ;(gt )t ; ;(gt ) + ;(gt ;(gt ))
tt t
= ; d;(gt)(gt ) ; ;(gtt ) ; 2;(gt ) +;(gt ;(gt )):
tt t
On the other hand we have by 2.5
R( gt)gt = d;( )(gt gt) ; d;(gt)( gt) ; ;( ;(gt gt)) +;(gt ;( gt)):
So satis es the Jacobi equation if and only if
= d;( )(gt gt) + 2;(gt ):
tt t
By plugging in formula 2.4 for ; and the formula for d; in the proof of 2.6,
the result follows.
The Riemannian manifold of all Riemannian metrics 17
4.3. Lemma. For any geodesic g(t) and for any k and ` 2 Tg(0)M(M) there
exists a unique vector eld (t) along g(t) which is a solution of the Jacobi
equation with (0) = k and (r@t )(0) = `.
Proof. From lemma 4.2 above we see that the Jacobi equation is pointwise
with respect to M, and that (t) satis es the Jacobi equation if and only if
at each x 2 M is a Jacobi eld of the associated nite dimensional manifold
treated in section 3. The result then follows from the properties of the nite
dimensional ordinary di erential equation involved.
4.4. Theorem. For any geodesic g(t) and for any k and ` 2 Tg(0)M(M)
there exists a unique Jacobi eld (t) along g(t) with initial values (0) = k
and (r@t )(0) = `.
In particular the solutions of the Jacobi equation are exactly the Jacobi
elds.
Proof. We have uniqueness since Jacobi elds satisfy the Jacobi equation and
by lemma 4.3. Now we prove existence. Let h = g0(0) and g0 = g(0). Since k
has compact support, there is an " > 0 such that for s 2 (;" ") the tensor eld
~
(s) = g0 + sk is still a Riemannian metric on M. Let ` := ` +;g0 (k h), and
~
then W(s) := h + s` is a vector eld along which satis es W(0) = h = g0(0)
and (r@s W)(0) = `. Now we consider (t s) := Exp ( s)(tW(s)), which is
de ned for all (t s) such that ( (t) tW(s)) belongs to the open set U of TM
de ned in 3.4.
Since h, k, and ` have all compact support we have: if g(t) is de ned on
[0 1), then " can be chosen so small that (t s) is de ned on [0 1) (;" ")
4
and if g(t) is de ned only on [0 ; ) then for each > 0 there is an " > 0
th
4
such that is de ned on [0 ; ; ) (;" ").
th
@
Then J(t) := j (t s) is a Jacobi eld for every geodesic segment and
0
@s
satis es
@ @ @
J(0) = j (0 s) = j Exp(0 ( s)) = j (s) = k
0 0 0
@s @s @s
@
(r@t J)(0) = r@t j (t ! j (t s))
7
0 0
@s
@
= r@s j (t ! j (t s)) = (r@s W)(0) = `
7
0 0
@t
where we used 2.1.
4.5. Since we will need it later we continue here with the explicit expression
~
of . As (s);1 W(s) = (Id + sK);1 (H + sL), where as usual K =(g0);1 k,
18 Olga Gil-Medrano, Peter W. Michor
~ ~
H =(g0);1 h, and L =(g0);1 `, from the expression in 3.3 of the exponential
mapping we get
(t s) = (s)eQ( t s) where
c(s)
~
Q(t s) = (a(t s)Id + b(t s)((Id + sK);1 (H + sL) ; Id)
n
~
where c(s) = tr((Id + sK);1 (H + sL)) and where a(t s) and b(t s) are given
by
;
2 1
a(t s) = log (4 + tc(s))2 + t2nd(s)
n 16
!
p
nd(s)t
4
p
b(t s) = arctg where
nd( s)
4 + tc(s)
c(s)2
d(s) = tr(( (s);1 W(s))2) = f (s) ; and
0
n
~
f (s) = tr ((Id + sK); 1 (H + sL))2 :
In fact b(t s) should be de ned with the same care as in the explicit formula
for the geodesics in 3.2. This is omitted here.
Before computing the Jacobi elds let us introduce some notation. For
every point in M the mapping (H K) ! tr(HK) is an inner product, thus
7
the quadrilinear mapping
T(H K L N) := tr(HL) tr(KN) ; tr(HN) tr(KL)
is an algebraic curvature tensor, [Kobayashi-Nomizu, I, page 198]. We will
also use
S(H K) := T(H K H K) = tr(H2) tr(K2) ; tr(HK)2:
Let P(t) := g(t);1g0(t) then it is easy to see from 3.2 that
1 4tr(H) + nt tr(H2)
2
P(t) = e; na( t) Id + H0 :
4n
;
1
1
2
We denote by P(t)? the -tensor eld of trace e; na( t) in the plane through
1
0, Id, and H0 which at each point is orthogonal to P(t) with respect to the
inner product tr(HK) from above.
The Riemannian manifold of all Riemannian metrics 19
4.6. Lemma. In the setting of 4. 5 we have
^ ^
@ tr(HL) T(L H Id H)
j Q(t s) = tP(t) + tP(t)?
0
@s tr(H2) tr(H2)
!
^
T(H Id L Id)
^
+ b(t) ; H0 + L0
S(H Id)
^ ~
where L := ;KH + L.
Proof. From the expression of Q(t s) we have
@ @ @
^
j Q(t s) = j a(t s)Id + j b(t s)H0 + b(t 0)L0 :
0 0 0
@s @s @s
Now,
@ 2 8tc0(0) + nt2f0(0)
j a(t s) =
0
@s n 16 + 8tc(0) + nt2f(0)
@ d0(0) 2 (4 + tc(0))td0(0) ; 2t2c0(0)d(0)
j b(t s) = ; b(t) + :
0
@s 2d(0) d(0) 16 +8tc(0) + nt2f (0)
From the de nitions of c, d, and f we have
^
c(0) = tr(H) c0(0) = tr(L)
2 1
d(0) = tr(H0 ) = S(H Id)
n
0
^
f(0) = tr(H2) f (0) = 2 tr(HL)
2
^
d0(0) = T(H Id L Id)
n
na( t)
2
and 16 + 8tc(0) + nt2f(0) = 16 e , and so we get
@ 1 1
^ ^
2
j a(t s) = e; na( t) (4t tr(L) + nt2 tr(HL))
0
@s 4n
^
@ T(H Id L Id)
j b(t s) = ; b(t)
0
@s S(H Id)
1
2 ;
e; na( t)t
^ ^
+ (4 + t tr(H))T(H Id L Id) ; t tr(L)S(H Id) :
4S(H Id)
If we collect all terms and compute a while we get the result.
20 Olga Gil-Medrano, Peter W. Michor
4.7. Theorem. Let g(t) be the geodesic in (M(M) G) starting from g0 in
the direction h 2 Tg0 M(M). For each k ` 2 Tg0 M(M) the Jacobi eld J(t)
along g(t) with initial conditions J(0) = k and (r@t J)(0) = ` is given by
^ ^ ^
tr(HL) tr(L) tr(H2) ; tr(LH) tr(H)
J(t) = t g0(t) + t (g0(t))?
tr(H2) tr(H2)
!
^
tr(H0L0)
^
+ b(t)g(t) ; H0 + L0
2
tr(H0 )
1
X
(; ad(b(t)H))m
^
+ g(t) (b(t)L) + k(g0);1 g(t)
(m + 1)!
m=1
where H =(g0);1 h, K =(g0);1 k, L =(g0);1 `,
^
0
L = ;KH + L +(g0);1 ;g (k h), and (g0(t))? = g(t)P(t)?.
Proof. By theorem 4.4 we have
@
J(t) = j (t s) and then
0
@s
@
= g0 j eQ( t s) + k eQ( t 0)
0
@s
!
1
X
(; ad(b(t)H))m @
= g0eQ( t 0) ( j Q(t s)) + k eQ( t 0):
0
(m + 1)! @s
m=0
The result now follows from lemma 4.6
References
DeWitt, B. S., Quantum theory of gravity. I. The canonical theory, Phys. Rev. 160 (5)
(1967), 1113{1148.
Ebin, D., The manifold of Riemannian metrics, Proc. Symp. Pure Math. AMS 15 (1970),
11-40.
Freed, D. S. Groisser, D., The basic geometry of the manifold o f Riemannian metrics and
of its quotient by the di eomorphism group, Michigan Math. J. 36 (1989), 323{344.
Fr Alfred Kriegl, Andreas, Linear spaces and di erentiation theory, Pure and Ap-
olicher,
plied Mathematics, J. Wiley, Chichester, 1988.
Greub, Werner Halperin, Steve Vanstone, Ray, Connections, Curvature, and Cohomology
I, Academic Press, New York and London, 1972.
Kainz, G. Michor, P. W., Natural transformations in di erential geometry, Czechoslovak
Math. J. 37 (1987), 584-607.
The Riemannian manifold of all Riemannian metrics 21
Kriegl, Andreas Michor, Peter W., A convenient setting for real analytic mappings, 52 p.,
to appear, Acta Mathematica (1990).
Kobayashi, S. Nomizu, K., Foundations of Di erential Geometry. Vol. I. , J. Wiley, 1963.
Michor, Peter W., Manifolds of di erentiable mappings, Shiva, Orpington, 1980.
Michor, P. W., Manifolds of smooth mappings IV: Theorem of De Rham, Cahiers Top.
Geo. Di . 24 (1983), 57{86.
Michor, Peter W., A convenient setting for di erential geometry and global analysis I, II,
Cahiers Topol. Geo. Di . 25 (1984), 63{109, 113{178..
Michor, P. W., Gauge theory for di eomorphism groups, Proceedings of the Conference on
Di erential Geometric Methods in Theoretical Physics, Como 1987, K. Bleuler and M.
Werner (eds.), Kluwer, Dordrecht, 1988, pp. 345{371.
Departamento de Geometr a y Topolog a, Facultad de Matematicas, Uni-
versidad de Valencia, 46100 Burjasot, Valencia, Spain.
 
Institut fur Mathematik, Universitat Wien, Strudlhofgasse 4, A-1090 Wien,
Austria.


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