Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 1

A diffeomorphism-invariant metric on the space of vector-valued one-forms

Pages: 141 – 183

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n1.a4

Authors

Martin Bauer (Department of Mathematics, Florida State University, Tallahassee, Fl., U.S.A.)

Eric Klassen (Department of Mathematics, Florida State University, Tallahassee, Fl., U.S.A.)

Stephen C. Preston (Department of Mathematics, Brooklyn College and the Graduate Center, City University of New York, N.Y., U.S.A.)

Zhe Su (Department of Mathematics, Florida State University, Tallahassee, Fl., U.S.A.)

Abstract

In this article we introduce a diffeomorphism-invariant Riemannian metric on the space of vector valued one-forms. The particular choice of metric is motivated by potential future applications in the field of functional data and shape analysis and by connections to the Ebin metric on the space of all Riemannian metrics. In the present work we calculate the geodesic equations and obtain an explicit formula for the solutions to the corresponding initial value problem. Using this we show that it is a geodesically and metrically incomplete space and study the existence of totally geodesic subspaces. Furthermore, we calculate the sectional curvature and observe that, depending on the dimension of the base manifold and the target space, it either has a semidefinite sign or admits both signs.

Keywords

space of Riemannian metrics, Ebin metric, sectional curvature, shape analysis

2010 Mathematics Subject Classification

Primary 58D15. Secondary 58B20.

Full Text (PDF format)

M. Bauer and Z. Su were partially supported by NSF-grant 1912037 (collaborative research in connection with NSF-grant 1912030). E. Klassen was partially supported by Simons Foundation, Collaboration Grant for Mathematicians, no. 317865 and S. C. Preston was partially supported by Simons Foundation, Collaboration Grant for Mathematicians, no. 318969.

Received 4 April 2020

Accepted 28 August 2020

Published 11 April 2021