AN EXISTENCE THEOREM FOR SUBMANIFOLDS OF HIGHER CODIMENSIONS

Pawe\l\ Witowicz

Key words: Affine immersion, type number.

1991 Math. Subject Classification: 53A15.

From the Introduction.

This paper concerns submanifolds of the standard affine space $\R^n$ of
codimension greater than one, equipped with affine connections.
In \Wit\ the notion of the type number of the second fundamental form
was adopted to the affine case and an equivalence theorem was proved, which
involved that concept. This time it is used to formulate and prove
a theorem about existence of an affine immersion of a manifold with
a torsion-free affine connection endowed with
objects which will be further the second fundamental form, the shape
operator and the normal connection for the obtained immersion.
The theorem requires less equations than a general existence
theorem formulated in \Dill.

In this paper some facts proved by Allendoerfer in the Riemannian case
(see \All) are generalized to the affine case. It is done in less
complicated way, without using local coordinates.
For comparison I also refer to a paper on existence theorems in
the case of hypersurfaces (\Opozda).