Induced generalized connections in vector subbundles

Irena Comi\'c

Abstract: Using special coordinate transformations we introduce subbundles and complementary subbundles of a vector bundle. The new results come from the fact that these bundles are considered together. In the former investigations as in Dragomir [5], Miron [7], Oproiu [8] the subbundle of the vector bundle was defined. In the relations between their tangent spaces the unit normal vectors of the subbundle were involved. Here, they are substituted by the tangent vectors of the complementary subbundle. \par The coordinates in the vector bundle $\xi=(E,\pi,M)$ are $(x^i,y^a)$ in the subbundle $\tilde\xi$ are $(u^\alpha,v^A)$ and in the complementary subbundle $\tilde{\tilde\xi}$ are $(\bar u^{\bar\alpha},\bar v^{\bar A})$. We need six types of indices. With respect to the special coordinate transformations (given by (1.1), (2.1) and (2.5)) the nonlinear connections $N_i^a(x,y)$, $N_{\alpha}^A(u,v)$ and $N_{\bar\alpha}^{\bar A}(\bar u,\bar v)$ are given. Using them, the adapted bases $B=\{\delta_i,\partial_a\}$ and $\widehat B =\{\delta_\alpha,\partial_A,\delta_{\bar\alpha},\partial_{\bar A}\}$ of $T(E)$ are constructed. The generalized connection $\nabla: T(E)\otimes T(E)\to T(E)$ in the basis $B$ has $2^3$ types and in the basis $\widehat B$ $4^3$ types of connection coefficients. The relations between these coefficients are given. These formulae are very general and have nice special cases. \par When the second fundamental forms of the subbundle and complementary subbundle are equal to zero, i.e.\ when the so called induced nonlinear connections $N_\alpha^A$ and $N_{\bar\alpha}^{\bar A}$ are used, then these relations are simpler ((3.4)$'$--(3.7)$'$). In this case we obtain that Miron's $d$-connection defined in $T(E)$ induces also $d$-connection in the tangent space of the subbundle, in $T(\widetilde E)$.

Classification (MSC2000): 53B25, 53B40

Full text of the article:

• Compressed PostScript file (83 kilobytes)
• PDF file (179 kilobytes)