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NONHOLONOMIC SEMI-RIEMANNIAN GEOMETRY WITH EXAMPLES ON H-TYPE GROUPS
ANNA KOROLKO AND IRINA MARKINA
Dedicated to Professor Fa´tima Silva Leite on the occasion of her 60th birthday.
Abstract. We introduce principal notions in nonholonomic semi-Riemannian geometry basing on ideas of sub-Riemannian and semi-Riemannian geometries. We study properties of Christo↵el symbols and the exponential map which are introduced analogously to sub-Riemannian geometry. We construct vari- ous examples of H(eisenberg)-type groups with left-invariant semi-Riemannian metrics and find their geodesics.
1. Introduction
The present article concerns the nonholonomic semi-Riemannian, or the sub - semi - Riemannian, geometry which is a new subject generalizing in a natural way sub-Riemannian and semi-Riemannian geometries. In this work the non-integrable smooth distribution (or subbundle of the tangent bundle to the manifold), which is often called horizontal, is equipped with a semi-Riemannian, i. e. a nondegenerate and indefinite, metric. The triple, consisting of a manifold, a horizontal subbundle and a semi-Riemannian metric on the horizontal subbundle, is called sub-semi- Riemannian manifold. In the case when the metric defined on the horizontal sub- bundle is positively definite, the sub-semi-Riemannian manifold is known as the sub-Riemannian manifold whose foundations can be found in [14, 17]. If the nonde- generate metric given on the horizontal subbundle is of index 1, the corresponding triple is called the sub-Lorentzian manifold (see, for example, [1, 3, 6, 7, 8, 10]). In the case, when the subbundle coincides with the whole tangent bundle, the corre- sponding manifold reduces to a Riemannian manifold provided the presence of a positively definite metric and to a semi-Riemannian manifold if the nondegenerate metric of an arbitrary index is given on the tangent bundle, see [15]. Therefore, non- holonomic manifolds with a bundle-type metric generalize classical Riemannian and semi-Riemannian manifolds, and these are abstract geometric objects for numerous
2000 Mathematics Subject Classification. 53C50, 83A05.
Key words and phrases. sub-semi-Riemannian manifold, nondegenerate metric, Hamiltonian system, exponential map, geodesic.
The work was supported by the grant of the Norwegian Research Council # 177355/V30, by the grant of the European Science Foundation Networking Programme HCAA, and by the NordForsk Research Network Programme # 080151.
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