SOME EXISTENCE AND NONEXISTENCE RESULTS OF ISOMETRIC IMMERSIONS OF RIEMANNIAN MANIFOLDS

Abstract

This paper investigates existence and non-existence of immersions of Riemannian manifolds. It discovers the lowest dimension of the Euclidean space into which the projective plane FP 2 is isometrically immersed, by the computation of the normal Euler class. For strictly hyperbolic immersion, a new obstruction involving signature or Kervaire semi-characteristic is found. As for the existence, it constructs a strictly hyperbolic immersion from the Klein bottle to the unit sphere S 3 (1), solving a question posed by Gromov.