Compatibility of Gauß maps with metrics

Abstract We give necessary and sufficient conditions on a smooth local map of a Riemannian manifold M m into the sphere S m to be the Gaus map of an isometric immersion u : M m → R n , n = m + 1 . We briefly discuss the case of general n as well.

[1]  C. B. Allendoerfer.  Rigidity for Spaces of Class Greater Than One , 1939 .

[2]  Kinetsu Abe,et al.  Isometric immersions with the same Gauss map , 1975 .

[3]  The Gauss equations and rigidity of isometric embeddings , 1983 .

[4]  M. Obata.  The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature , 1968 .

[5]  J. Nash.  The imbedding problem for Riemannian manifolds , 1956 .

[6]  M. Gromov,et al.  Embeddings and immersions in Riemannian geometry , 1970 .

[7]  Matthias Günther,et al.  Isometric Embeddings of Riemannian Manifolds , 2010 .

[8]  M. Gromov,et al.  Partial Differential Relations , 1986 .

[9]  Robert Everist Greene,et al.  Isometric embeddings of Riemannian and pseudo-Riemannian manifolds , 1970 .

[10]  E. Cartan,et al.  Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien , 1928 .

[11]  Jia-Xing Hong,et al.  Isometric Embedding of Riemannian Manifolds in Euclidean Spaces , 2006 .

[12]  T. Vlachos.  Isometric deformations of surfaces preserving the third fundamental form , 2008 .

[13]  Gauss maps and symmetric spaces , 2008 .

[14]  Matthias Günthier,et al.  On the perturbation problem associated to isometric embeddings of Riemannian manifolds , 1989 .

[15]  Detlef Gromoll,et al.  Real Kaehler submanifolds and uniqueness of the Gauss map , 1985 .

[16]  Detlef Gromoll,et al.  Euclidean hypersurfaces with isometric Gauss maps , 1986 .

[17]  B. Andrews,et al.  Notes on the isometric embedding problem and the Nash-Moser implicit function theorem , 2002 .