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Abstract. Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type submanifolds was published in [40]. A detailed survey of the results, up to 1996, on this subject was given by the author in [48]. Recently, the study of finite type submanifolds, in particular, of biharmonic submanifolds, have received a growing attention with many progresses since the beginning of this century. In this article, we provide a detailed account of recent development on the problems and conjectures listed in [40].

A slant immersion is defined as an isometric immersion from a Riemannnian manifold into an almost Hermitian manifold with constant Wirtinger angle. In this article we give some fundamental results concerning slant immersions. Several results on slant surfaces in ℂ2 are also proved.

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In this paper we study the following problem: To what extent does the type of the Gauss map of a submanifold of W 1 determine the submanifold? Several results in this respect are obtained.In particular, submanifolds with 2-type Gauss map are characterized.Surfaces with ,2-type Gauss map and minimal surfaces of S with 2-type Gauss map are completely classified. Some applications are also given. Introduction.A compact submanifold M of a Euclidean m-space ET is said to be of finite type if the immersion x of M in E" 1 can be expressed as a finite sum of ^-valued eigenfuctions of the Laplacian A of M ,acting on E -valued functions. Minimal submanifolds of a hypersphereReceived 12 March 1986. This work was done while the first author was a C.N.R. visiting professor at the University of Rome. He would like to take this opportunity to express his thanks to the Consiglio Nazionale delle Ricerche of Italy for the invitation and financial support, and also to express his many thanks to his colleagues" at Rome for their hospitality. G(n,m) is canonically imbedded in hu=K,N=(),the notion of finite-type Gauss map is naturally defined.The main purpose of this paper is to study the following problem: To what extent does the type of the Gauss map of a submanifold of ET determine the submanifold?For closed curves in E , the type of a curve in E coincides with that of its Gauss map (Proposition 3.1). In contrast, for submanifolds of dimension £ 2 , the two notions are different.A well-known result of Takahashi says that a compact submanifold of E is of 2-type if and only if it is a minimal submanifold of a hypersphere. In Section 4 we study the following problem: Which submanifolds of E have i-type Gauss map? In this respect, we obtain a chacterization theorem for submanifolds with i-type Gauss map. This result is then applied to obtain some classification theorems of such submanifolds.In Section 5, we show that a standard isometric immersion of an ordinary 2-sphere has 2-type Gauss map if and only if it is not the first standard imbedding. The complete classification of flat minimal tori in with 2-type Gauss map is given in Section 6. In the last section, wegive the complete classification of minimal surfaces of S Preliminaries.Let The following result is known (see [5,7] Gauss Map.Let V be an oriented w-plane in E" 1 . Denote by e ,..

The warped product N 1 × f N 2 of two Riemannian manifolds (N 1 , g 1 ) and (N 2 , g 2 ) is the product manifold N 1 × N 2 equipped with the warped product metric g = g 1 + f 2 g 2 , where f is a positive function on N 1 . The notion of warped product manifolds is one of the most fruitful generalizations of Riemannian products. Such notion plays very important roles in differential geometry as well as in physics, especially in general relativity. Warped product manifolds have been studied for a long period of time. In contrast, the study of warped product submanifolds was only initiated around the beginning of this century in a series of articles [36,39,40,43]. Since then the study of warped product submanifolds has become a very active research subject.In this article we survey important results on warped product submanifolds in various ambient manifolds. It is the author's hope that this survey article will provide a good introduction on the theory of warped product submanifolds as well as a useful reference for further research on this vibrant research subject.