A remark on structures in tangent bundles

A. P. Shirokov

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 5 (1974), pp. 311-318.

Résumé

In the theory of tangent bundle $T^r(M)$ over a differentiable manifold $M$ of class $C^\omega$ а structure arises which is determined with the help of aglebra $\mathbf R(\varepsilon)$. This aglebra is the result of elements $\mathbf 1$ et $\varepsilon$, where $\varepsilon^{r+1}=0$. With the help of this algebra it is simple to build the lifts of tensor fields from $M$ in $T^r(M)$. As an example a group of motions of Euclidean space $R_3$ is considered which can be interpretated both as the real model of elliptic space $S_3(\varepsilon)$ over a algebra of dual numbers and as the tangent bundle $T(S_3)$.