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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 finite dimensional manifold, the space M = M(M ) of all Riemannian metrics on it can be endowed with a structure of an infinite dimensional smooth manifold modeled on the space D(S2T \Lambda M ) of symmetric \Gamma 02\Delta -tensor fields with compact support, in the sense of [Michor, 1980]. The tangent bundle of M is T M = C1(S2+T \Lambda M ) \Theta D

(S2T \Lambda M ) and a smooth Riemannian metric can be defined by

Gg(h; k) = Z

M tr(g

\Gamma 1hg\Gamma 1k) vol(g):

In this paper we study the geometry of (M; G) by using the ideas developed in [Michor, 1980].

With that differentiable 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.

Christoffel symbol and curvature turn out to be pointwise in M and so, although the mappings involved in the definition of the Ricci tensor and the scalar curvature have no trace, in our case we can define 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 definition of the

The first 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'ia de Cultura, Educaci'on y Ciencia, Generalidad Valenciana.

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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 diffeomorphism 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 diffeomorphism, using [Kriegl-Michor, 1990]. We think that this exponential mapping will be a very powerful tool for further investigations of the stratification of orbit space of M under the diffeomorphism group, and also the stratification of principal connections modulo the gauge group.

In section 4 Jacobi fields of an infinite dimensional Riemannian manifold are defined as the infinitesimal geodesic variations and we show that they must satisfy the Jacobi Equation. For the manifold (M; G) the existence of Jacobi fields, 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 finally give the expresion of the Jacobi fields.

For fixed x 2 M , there exists a family of homothetic Riemannian metrics in the finite dimensional manifold S2+T \Lambda x 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 infinite dimensional manifold.

Metrics on S2+T \Lambda x M for three dimensional manifolds M which are similar to ours but have different signatures were considered by [DeWitt, 1967]. He computed the curvature and the geodesics and gave some ideas on how to use them to determine the distance between two 3-geometries, but without considering explicitly the infinite dimensional manifold of all Riemannian metrics on a given manifold.