45. Manifolds of Riemannian Metrics

The usual metric on the space of all Riemannian metrics was considered by [Ebin,

1970], who used it to construct a slice for the action of the group of di¬eomorphism

on the space of all metrics. It was then reconsidered by [Freed, Groisser, 1989],

and by [Gil-Medrano, Michor, 1991] for noncompact M . The results in this section

are largely taken from the last paper and from [Gil-Medrano, Michor, Neuwirther,

1992].

45.1. Bilinear structures. Throughout this section let M be a smooth second

countable ¬nite dimensional manifold. Let —2 T — M denote the vector bundle of all

0 —

2 -tensors on M , which we canonically identify with the bundle L(T M, T M ). Let

GL(T M, T — M ) denote the non degenerate ones. For any b : Tx M ’ Tx M we let

—

b—

——

’—

the transposed be given by bt : Tx M ’ Tx M ’ Tx M . As a bilinear structure b

is skew symmetric if and only if bt = ’b, and b is symmetric if and only if bt = b.

In the latter case a frame (ej ) of Tx M can be chosen in such a way that in the dual

—

frame (ej ) of Tx M we have

b = e1 — e1 + · · · + ep — ep ’ ep+1 — ep+1 ’ ep+q — ep+q ;

b has signature (p, q) and is non degenerate if and only if p + q = n, the dimension

of M . In this case, q alone will be called the signature.

A section b ∈ C ∞ (GL(T M, T — M )) will be called a non degenerate bilinear struc-

ture on M , and we will denote the space of all such structures by B(M ) = B :=

C ∞ (GL(T M, T — M )). It is open in the space of sections C ∞ (L(T M, T — M )) for the

Whitney C ∞ -topology, in which the latter space is, however, not a topological vec-

tor space, as explained in detail in (41.13). The space Bc := Cc (L(T M, T — M )) of

∞

sections with compact support is the largest topological vector space contained in

the topological group (C ∞ (L(T M, T — M )), +), and the trace of the Whitney C ∞ -

topology on it induces the convenient vector space structure described in (30.4).

So we declare the path components of B = C ∞ (GL(T M, T — M )) for the Whitney

C ∞ -topology also to be open; these are open in a¬ne subspaces of the form b + Bc

for some b ∈ B and we equip them with the translates of the c∞ -topology on Bc .

The resulting topology is ¬ner than the Whitney topology and will be called the

natural topology, similar as in (42.1).

45.1

488 Chapter IX. Manifolds of mappings 45.3

45.2. The metrics. The tangent bundle of the space B = C ∞ (GL(T M, T — M ))

of bilinear structures is T B = B — Bc = C ∞ (GL(T M, T — M )) — Cc (L(T M, T — M )).

∞

Then b ∈ B induces two ¬berwise bilinear forms on L(T M, T — M ) which are given by

(h, k) ’ tr(b’1 hb’1 k) and (h, k) ’ tr(b’1 h) tr(b’1 k). We split each endomorphism

tr(H)

H = b’1 h : T M ’ T M into its trace free part H0 := H ’ dim M Id and its

trace part which simpli¬es some formulas later on. Thus, we have tr(b’1 hb’1 k) =

tr((b’1 h)0 (b’1 k)0 )+ dim M tr(b’1 h) tr(b’1 k). The structure b also induces a volume

1

density on the base manifold M by the local formula

| det(bij )| |dx1 § · · · § dxn |.

vol(b) =

For each real ± we have a smooth symmetric bilinear form on B, given by

(tr((b’1 h)0 (b’1 k)0 ) + ± tr(b’1 h) tr(b’1 k)) vol(b).

G± (h, k) =

b

M

It is invariant under the action of the di¬eomorphism group Di¬(M ) on the space B

of bilinear structures. The integral is de¬ned since h and k have compact supports.

For n = dim M we have

1/n

tr(b’1 hb’1 k) vol(b),

Gb (h, k) := Gb (h, k) =

M

which for positive de¬nite b is the usual metric on the space of all Riemannian

metrics. We will see below in (45.3) that for ± = 0 it is weakly non degenerate,

i.e. G± de¬nes a linear injective mapping from the tangent space Tb B = Bc =

b

Cc (L(T M, T — M )) into its dual Cc (L(T M, T — M )) , the space of distributional

∞ ∞

densities with values in the dual bundle. This linear mapping is, however, never

surjective. So we have a one parameter family of pseudo Riemannian metrics on the

in¬nite dimensional space B which obviously is smooth in all appearing variables.

45.3. Lemma. For h, k ∈ Tb B we have

tr(b’1 h)b, k),

G± (h, k) = Gb (h + ±n’1

b n

tr(b’1 h)b, k), if ± = 0,

Gb (h, k) = G± (h ’ ±n’1

b ±n2

where n = dim M . The pseudo Riemannian metric G± is weakly non degenerate

for all ± = 0.

Proof. The ¬rst equation is an obvious reformulation of the de¬nition, the sec-

ond follows since h ’ h ’ ±n’1 tr(b’1 h)b is the inverse of the transform h ’

±n2

h + n tr(b h)b. Since tr(b’1 hx (b’1 hx )t,g ) > 0 if hx = 0, where t,g is the

’1

±n’1

x x

transposed of a linear mapping with respect to an arbitrary ¬xed Riemannian met-

ric g, we have

Gb (h, b(b’1 h)t,g ) = tr(b’1 h(b’1 h)t,g ) vol(b) > 0

M

if h = 0. So G is weakly non degenerate, and by the second equation G± is weakly

non degenerate for ± = 0.

45.3

45.6 45. Manifolds of Riemannian metrics 489

45.4. Remark. Since G± is only a weak pseudo 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± (ξ, ±

=ξG± (·, ζ) + ·G± (ζ, ξ) ’ ζG± (ξ, ·)

· ζ)

+ G± ([ξ, ·], ζ) + G± ([·, ζ], ξ) ’ G± ([ζ, ξ], ·),

which a priori gives only uniqueness but not existence of the Levi Civita covariant

derivative. We will show that it exists and we use it in the form explained in (37.28).

—

45.5. Lemma. For x ∈ M the pseudo metric on GL(Tx M, Tx M ) given by

γbx (hx , kx ) := tr((b’1 hx )0 (b’1 kx )0 ) + ± tr(b’1 hx ) tr(b’1 kx )

±

x x x x

n(n’1)

has signature (the number of negative eigenvalues) for ± > 0 and has sig-

2

nature ( n(n’1) + 1) for ± < 0.

2

Proof. In the framing H = b’1 hx and K = b’1 kx we have to determine the

x x

signature of the symmetric bilinear form (H, K) ’ tr(H0 K0 )+± tr(H) tr(K). Since

the signature is constant on connected components we have to determine it only

1 1

for ± = n and ± = n ’ 1.

1

For ± = n we note ¬rst that on the space of matrices (H, K) ’ tr(HK t ) is positive

de¬nite, and since the linear isomorphism K ’ K t has the space of symmetric

matrices as eigenspace for the eigenvalue 1 and the space of skew symmetric matrices

as eigenspace for the eigenvalue ’1, we conclude that the signature is n(n’1) in

2

this case.

1

For ± = n ’ 1 we proceed as follows: On the space of matrices with zeros on

the main diagonal the signature of (H, K) ’ tr(HK) is n(n’1) by the argument

2

above and the form (H, K) ’ ’ tr(H) tr(K) vanishes. On the space of diagonal

matrices which we identify with Rn the whole bilinear form is given by x, y =

ii i i n

ix y ’( i x )( i y ). Let (ei ) denote the standard basis of R , and put a1 :=

1

n (e1 + · · · + en ) and

1

(e1 + · · · + ei’1 ’ (i ’ 1)ei )

ai :=

1)2

i ’ 1 + (i ’

1

for i > 1. Then a1 , a1 = ’1 + n, and for i > 1 we get ai , aj = δi,j . So the

signature is 1 in this case.

45.6. Let t ’ b(t) be a smooth curve in B. So b : R — M ’ GL(T M, T — M ) is

smooth, and by the choice of the topology on B made in (45.1) the curve b(t) varies

only in a compact subset of M , locally in t, by (30.9). Then its energy is given by

a2

a2

G± (bt , bt )dt

1

Ea1 (b) := b

2

a1

a2

tr((b’1 bt )0 (b’1 bt )0 ) + ± tr(b’1 bt )2 vol(b) dt,

1

= 2

a1 M

45.6

490 Chapter IX. Manifolds of mappings 45.8

‚

where bt = ‚t b(t).

Now we consider a variation of this curve, so we assume that (t, s) ’ b(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 (45.1). Note that b(t, 0)

is the original b(t) above.

45.7. Lemma. In the setting of (45.6), we have the ¬rst variation formula

, s)) = G± (bt , bs )|t=a1 +

± a1

‚

‚s |0 E(G )a0 (b( t=a0

b

a1

1 1

G(’btt + bt b’1 bt +tr(b’1 bt b’1 bt )b ’ tr(b’1 bt )bt +

+

4 2

a0

1

+ ± (’ tr(b’1 btt ) ’ tr(b’1 bt )2 + tr(b’1 bt b’1 bt ))b, bs ) dt =

4

= G± (bt , bs )|t=a1 +

t=a0

b

a1

1 1

tr(b’1 b’1 bt )b+

G± (’btt + bt b’1 bt ’ tr(b’1 bt )bt +

+ t

2 4±n

a0

±n ’ 1

tr(b’1 bt )2 b, bs ) dt.

+ 2

4±n

‚

Proof. We may interchange ‚s |0 with the ¬rst integral describing the energy in

(45.6) since this is ¬nite dimensional analysis, and we may interchange it with the

second one, since M is a continuous 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 continuous, d(vol)(b)h = 2 tr(b’1 h) vol(b), and that d(( )’1 )— (b)h =

1

’b’1 hb’1 , and partial integration.

45.8. The geodesic equation. By lemma (45.7), the curve t ’ b(t) is a geodesic

if and only if we have

±n ’ 1

1 1

btt = bt b’1 bt ’ tr(b’1 bt )bt + tr(b’1 bt b’1 bt )b + tr(b’1 bt )2 b.

4±n2

2 4±n

= “b (bt , bt ),

where the G± -Christo¬el symbol “± : B — Bc — Bc ’ Bc is given by symmetrization

1 ’1 1 1 1

hb k + kb’1 h ’ tr(b’1 h)k ’ tr(b’1 k)h+

“± (h, k) =

b

2 2 4 4

±n ’ 1

1

tr(b’1 hb’1 k)b + tr(b’1 h) tr(b’1 k)b.

+

4±n2

4±n

The sign of “± is chosen in such a way that the horizontal subspace of T 2 B is

parameterized by (x, y; z, “x (y, z)). If instead of the obvious framing we use T B =

(b, h) ’ (b, b’1 h) =: (b, H) ∈ {b} — Cc (L(T M, T M )), the Christo¬el

∞

B — Bc

symbol looks like

1 1 1

±

(HK + KH) ’ tr(H)K ’ tr(K)H

“b (H, K) =

2 4 4

±n ’ 1

1

+ tr(HK) Id + tr(H) tr(K),

4±n2

4±n

45.8

45.11 45. Manifolds of Riemannian metrics 491

and the G± -geodesic equation for B(t) := b’1 bt becomes

±n ’ 1

1 1

’1

tr(B)2 Id .

‚

tr(BB) Id ’ tr(B)B +

Bt = ‚t (b bt ) =

4±n2

4±n 2

45.9. The curvature. For another manifold N , for vector ¬elds X, Y ∈ X(N )

and a vector ¬eld s : N ’ T M along f : N ’ M we have

])s = (K —¦ T K ’ K —¦ T K —¦ κT M ) —¦ T 2 s —¦ T X —¦ Y,

R(X, Y )s = ( ’[ X, Y

[X,Y ]

where K : T T M ’ M is the connector (37.28), κT M is the canonical ¬‚ip T T T M ’

T T T M (29.10), and where the second formula in local coordinates reduces to the

usual formula

R(h, k) = d“(h)(k, ) ’ d“(k)(h, ) ’ “(h, “(k, )) + “(k, “(h, )),

(1)

see [Kainz, Michor, 1987] or [Kol´ˇ, Michor, Slovak, 1993, 37.15].

ar

45.10. Theorem. The curvature for the pseudo Riemannian metric G± on the

manifold B of all non degenerate bilinear structures is given by

1 1

b’1 R± (h, k)l = [[H, K], L] + (’ tr(HL)K + tr(KL)H)+

b

4 16±

4±n ’ 3±n2 + 4n ’ 4

(tr(H) tr(L)K ’ tr(K) tr(L)H)+

+

16±n2

4±2 n2 ’ 4±n + ±n2 + 3

(tr(HL) tr(K) Id ’ tr(KL) tr(H) Id),

+

16±n2

where H = b’1 h, K = b’1 k and L = b’1 l.

Proof. This is a long but direct computation starting from (45.9.1).

The geodesic equation can be solved explicitly, and we have

45.11. Theorem. Let b0 ∈ B and h ∈ Tb0 B = Bc . Then the geodesic for the

metric G± in B starting at b0 in the direction of h is the curve

exp±0 (th) = b0 e(a(t) Id +b(t)H0 ) ,

b

where H0 is the traceless part of H := (b0 )’1 h (i.e. H0 = H ’ tr(H) Id), and where

n

∞

a(t) = a±,H (t) and b(t) = b±,H (t) in C (M, R) are de¬ned as follows:

’1

2 t 2±

2 2

a±,H (t) = log (1 + tr(H)) + t tr(H0 ) ,

n 4 16

±

2

t ±’1 tr(H0 )

4

for ±’1 tr(H0 ) > 0

2

±’1 tr(H 2 ) arctan

4 + t tr(H)

0

2

t ’±’1 tr(H0 )

4

b±,H (t) = for ±’1 tr(H0 ) < 0

2

artanh

4 + t tr(H)

2

’±’1 tr(H0 )

t

2

for tr(H0 ) = 0.

t

1 + 4 tr(H)

45.11

492 Chapter IX. Manifolds of mappings 45.11

Here arctan is taken to have values in (’ π , π ) for the points of the base manifold,

22

where tr(H) ≥ 0, and on a point where tr(H) < 0 we de¬ne

arctan in [0, π ) for t ∈ [0, ’ tr(H) )

4

±

2

2

t ±’1 tr(H0 )

π 4

for t = ’ tr(H)

arctan = 2

4 + t tr(H)