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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)
##### MSC:
 53C05 Connections (general theory) 53B99 Local differential geometry