zbMATH — the first resource for mathematics

Cross-section submanifolds in vector bundles. (English) Zbl 0790.53026
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 23-31 (1992).
Let \((E,\pi,M)\) be a vector bundle, \(\pi(E)=M\), \(\dim E=n+m\), \(S: M\to E\) a \(C^ \infty\)-section of \(\pi\) and \(B\) the tangent map \(S^ T\). Thus (1) \(TE\mid_{S(M)}=B(TM)\oplus VE\) holds, \(V\) being the vertical distribution on \(E\). The author studies the embedding \(S\) by using (1), when \(E\) is endowed with some geometrical objects: (a) a nonlinear connection \(N\); (b) \(N\) and a linear \(d\)-connection \(D\); (c) \(N\), \(D\) and a metrical structure \(G\). For each of these cases the Gauss-Weingarten formulas and Gauss-Codazzi equations are studied, too.
For the entire collection see [Zbl 0764.00002].
Reviewer: R.Miron (Iaşi)
53C05 Connections (general theory)
53B99 Local differential geometry
PDF BibTeX Cite