Problems On Ideal Immersions

An isometric immersed submanifold which receives the least possible amount of tension from the surrounding space at each point is called an ideal immersion. In this paper, we study a kind of ideal immersions which were introduced by Chen.Einstein, conformally flat, semi-symmetric and semi-parallel immersions of a real-space-form are considered in Chapter 1. First, a large class of ideal Einstein and conformally flat immersions are proved to be totally geodesic, which generalizes a theorem of Dillen on the simplest ideal immersion. Second, we completely classify ideal immersed hypersurfaces of an Euclidean space. Moreover, ideal hypersurfaces in a nontrivial real-space-form are proved to be isoparametric submanifolds with at most two distinct principal curvatures. At last, we also give the classifying theorem of ideal semi-parallel immersiohs in an Euclidean space with flat normal bundles.Ideal immersions in complex-space-forms are studied in Chapter 2. Chen proved that each ideal Lagrangian immersion is minimal. And we show that if such a immersion is also semi-parallel, it must be totally geodesic. Moreover the same results hold for semi-symmetric submanifolds in a nontrivial complex-space-form. Ideal Kaehlerian slant immersions in a complex space form are also proved to be minimal, which is a natural generalization of Chen's result. At the end of this Chapter, we prove several nonexistent theorems for proper slant immersions, which also generalizes Chen and Tazawa's theorem of higher dimension.In Chapter 3, we study a class of contact metric manifolds------contact metric(k,u)-spaces which were first introduced by Blair. Boeckx proves that these spaces have good properties, and we classify a large class of their submanifolds. This classification contributes to studying the structure of the underlying space, which is the purpose of many scholars.