A Case for Curvature: the Unit Tangent Bundle

Abstract

In the scientific work of L. Vanhecke, the notion of curvature is never more than a step away, if not studied explicitly. This is only right, since, in the words of R. Osserman, “curvature is the central concept (in differential geometry and, more in particular, in Riemannian geometry), distinguishing the geometrical core of the subject from those aspects that are analytic, algebraic or topological”. The reason for this can be seen as follows:

• if we equip a differentiable manifold M with a metric g, then its curvature is completely determined. If the metric g has nice properties (e.g., a large group of isometries), then this is reflected in a ‘nice’ curvature;

• conversely, we can often deduce information about the metric from special properties of the curvature. In some cases, knowledge about the curvature even suffices to completely determine the metric (at least locally). Locally symmetric spaces are the prime example here: they are distinguished from non-symmetric spaces by their parallel curvature and, starting from the curvature, one can reconstruct the manifold and its metric (locally).

The curvature information is contained in the Riemannian curvature tensor R. This is an analytic object, a (0, 4)-tensor which is not easy to handle, in general, despite its many symmetries. It is often very difficult to extract the geometrical information which is, as it were, encoded within. For this reason, the famous geometer M. Gromov calls the curvature tensor “a little monster of multilinear algebra whose full geometric meaning remains obscure”.