Showing papers on "Unit tangent bundle published in 1971"

Curvature of the Induced Riemannian Metric on the Tangent Bundle of a Riemannian Manifold.

Abstract: If M is an ra-dimensional differentiable manifold (n £ N) with the Riemannian metric g, then the tangent b ndle TM of M admits a canonical Riemannian metric Tg (see [1], [2]). In other words, a metric connection v on M induces, in a canonical way, a metric connection v on TM. Further, A. J. Ledger and K. Yano ([3], [4]) found a different construction joining to any linear connection v on M a linear connection v on TM. The basic result of [4] says that the space (M, v) is locally Symmetrie if and only if the space (T M, v) is locally Symmetrie, In the present paper we compute the curvature tensor of the Riemannian space (TM, Tg) corresponding to (If, g). We deduce, in contrast to the Yano-Ledger's theory, the following result: The space (TM, Tg) is never locally Symmetrie unless (M, g) is locally euclidean.

Some cohomology classes in principal fiber bundles and their application to riemannian geometry.

Abstract: We define some new global invariants of a fiber bundle with a connection. They are cohomology classes in the principal fiber bundle that are defined when certain characteristic curvature forms vanish. In the case of the principal tangent bundle of a riemannian manifold, they are invariant under a conformal transformation of the metric. They give necessary conditions for conformal immersion of a riemannian manifold in euclidean space.

Vector bundle valued harmonic forms and immersions of Riemannian manifolds

Abstract: The purpose of this paper is to discuss an application of the theory of vector bundle valued harmonic forms on a Riemannian manifold to the study of immersions. Let M be a Riemannian manifold and E a Riemannian vector bundle over M. Then we can define in a natural way the Laplacian Π operating on ^-valued differential forms and we can express the scalar product <•#, 0>, where θ is an £"-valued />-form, in terms of curvature and covariant differentials. Moreover, if M is compact, we obtain, by integrating over M, a formula analogous to Bochner's for ordinary (i.e. real valued) differential forms. Let/be an immersion of M into a Riemannian manifold M'. We may regard the second fundamental form a of (M,f) as a Horn (Γ(M), 7V(M))-valued 1-form. Assuming that M' is of constant sectional curvature, we shall prove that the second fundamental form a is harmonic, i.e. \^Ja = 0, if the mean curvature normal of (M, f) is parallel. In particular, if the immersion/is a minimal immersion, then a is harmonic. Conversely, if M is compact and if a is harmonic, then the mean curvature normal is parallel. We obtain from this result together with the formula of Bochner type the results of Simons [5], Chern [1], Nomizu-Smyth [4] and Erbacher [2] proved by them in different ways. In a future paper we shall discuss the case where M is a Kahler manifold.