Codazzi fields on surfaces immersed in Euclidean 4-space
Consider a surface \(M\) immersed in \(\mathbb R^4\) and a vector field \(\nu\) in the normal bundle \(NM\). The authors deal with the question, under what conditions there exists an isometric immersion of \(M\) in \(\mathbb R^3\) so that there is a vector bundle isomorphism between the sub-bundle determined by \(\nu\) in \(NM\) and the normal bundle of the surface \(M'\) in \(\mathbb R^3\), taking the second fundamental form \(II_\nu\) of \(M\) to the second fundamental form of \(M'\). In the paper necessary and sufficient conditions for the normal field \(\nu\) are determined in order to get an affirmative answer to the above question. Furthermore geometric properties are deduced the surface \(M\) must fulfill in order to admit an isometric immersion in \(\mathbb R^3\) with prescribed second fundamental form.
scientific article; zbMATH DE number 5509685
scientific article