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.

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

Publisher Summary Using the notion of complete lift it is shown that G structures as a pseudo-Riemannian structure, an almost complex structure and a symplectic structure on M induce similar structures on the tangent bundle T(M). An unexpected but perhaps more interesting result is that each pseudo-Riemannian (resp. affine) symmetric space structure on M induces a pseudo-Riemannian (resp. affine) symmetric space structure on T(M). This suggests a method of producing a large class of affine symmetric spaces. The chapter considers A be a local algebra of the form A=R+I where R is the field of real numbers and I is the maximal ideal of A such that dim I k =0 for some k. A successful generalization of the theory to A(M) furnishs a useful tool for the differential geometry of higher order contact and yield a large number of affine symmetric spaces.

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1260541201

Prolongations of tensor fields and connections to tangent bundles I – General theory –

In our previous paper [3] we introduced the notion of complete lift of an afl\"ine connection. Let M be a manifold T(M) its tangent bundle space. Then every affine connection 17 of M induces in a natural manner an affine connection, called the complete lift 17c of 17, of the manifold T(M). We shall show in this paper that the linear holonomy group L(I7c) of the connection UC coincides with the tangent group T(~(V)) of the linear holonomy group P(P) of the connection 17, i. e., ~(Pc) = T(P(G)).

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Prolongations of tensor fields and connections to tangent bundles III

Publisher Summary This chapter considers various methods by which certain types of tensor fields in M can be extended to c T (M) to give useful information about the relationships between the structures of the two manifolds. The chapter focuses on complete lifts of vector fields, tensor fields of type (1, 1) and skewsymmetric tensor fields of type (1, 2). In each of these cases, the complete lift is defined to be a tensor field of the same type as the original. In general, the vertical lift of a tensor field does not have the same type as the original; nevertheless the construction is a useful one. The methods enable to examine the structure of C T( M ) in relation to that of M. In particular, it is shown how almost complex and similar structures on M can be extended to C T(M). Lifts of affine connections in M is also examined, using the idea of a Riemann extension.

/pdf/vertical-and-complete-lifts-from-a-manifold-to-its-cotangent-3vcskros10.pdf

Vertical and complete lifts from a manifold to its cotangent bundle

Structures on differentiable manifolds have been very intensively studied in recent years both in differential topology as well as in differential geometry. Along with almost complex structures a systematic examination was begun of almost contact, affinor, symplectic, quaternion, and many other structures, a simple listing of which would take up considerable space. Not all of these structures are of equal value, but there is no doubt that the study of manifolds with various structures is of interest from both the global as well as the local point of view; moreover, for certain structures the possibility was outlined for applying them to various aspects of geometry and mechanics.

A. J. Ledger

Papers

Linear Connections on Tangent Bundles

The Tangent Bundle of a Locally Symmetric Space