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The Foundations Of Differential Geometry

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].

Some open problems and conjectures on submanifolds of finite type: recent development

In this article we study surfaces in $ {\Bbb S}^2\times {\Bbb R} $ for which the unit normal makes a constant angle with the $ {\Bbb R} $-direction. We give a complete classification for surfaces satisfying this simple geometric condition.

Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

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In this paper we classify constant angle surfaces in $\H^2\times\R$, where $\H^2$ is the hyperbolic plane.

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Constant Angle Surfaces in $\H^2\times \R$

Let x : M → E m be an isometric immersion from a Riemannian n-manifold into a Euclidean m-space. Denote by Δ and x → the Laplace operator and the position vector of M, respectively. Then M is called biharmonic if Δ 2 x → = 0 . The following Chenʼs Biharmonic Conjecture made in 1991 is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper we prove that the biharmonic conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces of a Euclidean space of arbitrary dimension.

Biharmonic ideal hypersurfaces in Euclidean spaces

We explicitly determine all magnetic curves corresponding to the Killing magnetic fields on the 3-dimensional Euclidean space.

Magnetic curves corresponding to killing magnetic fields in e 3

The classification of Killing magnetic curves in S2×R

We classify all surfaces in ℍ2 × ℝ for which the unit normal makes a constant angle with the ℝ-direction. Here ℍ2 is the hyperbolic plane.

Marian Ioan Munteanu

Papers

Constant Angle Surfaces in $\H^2\times \R$

Biharmonic ideal hypersurfaces in Euclidean spaces

Magnetic curves corresponding to killing magnetic fields in e 3

The classification of Killing magnetic curves in S2×R

Constant angle surfaces in ℍ2 × ℝ