TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics—Volume 1) By Bang-Yen Chen: pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley & Sons Ltd, 1984.)

The theory of Lie derivatives and its applications

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

Submanifolds of Finite Type

The notion of being totally umbilic is considered for non-degenerate and degenerate submanifolds of semi-Riemannian manifolds. After some remarks on the general case, timelike and lightlike totally umbilic submanifolds of Lorentzian manifolds are discussed, along with their physical interpretation in view of general relativity. In particular, the mathematical notion of totally umbilic submanifolds is linked to the notions of photon surfaces and of null strings which have been used in the physics literature.

On totally umbilic submanifolds of semi-Riemannian manifolds

Some characterizations of spheres and elliptic paraboloids II

The helicoidal surfaces with pointwise 1-type or harmonic gauss map in Euclidean 3-space are studied. The notion of pointwise 1- type Gauss map is a generalization of usual sense of 1-type Gauss map. In particular, we prove that an ordinary helicoid is the only genuine he- licoidal surface of polynomial kind with pointwise 1-type Gauss map of the flrst kind and a right cone is the only rational helicoidal surface with pointwise 1-type Gauss map of the second kind. Also, we give a charac- terization of rational helicoidal surface with harmonic or pointwise 1-type Gauss map.

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Helicoidal surfaces with pointwise 1-type gauss map

In this article, we study ruled surfaces in a Lorentz-Minkowski
space, which has finite type immersion. We give a complete classification of
ruled surfaces of finite type immersion into a Lorentz-Minkowski space with
arbitrary codimension.

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Finite type ruled surfaces in lorentz-minkowski space

. Archimedes knew that the area between a parabola and anychord AB on the parabola is four thirds of the area of triangle ∆ABPwhere P is the point on the parabola at which the tangent is parallel toAB. We consider whether this property (and similar ones) characterizesparabolas. We present ﬁve conditions which are necessary and suﬃcientfor a strictly convex curve in the plane to be a parabola. 1. IntroductionA parabola is the set of points in the plane which are equidistant from apoint F called the focus and a line l called the directrix. Archimedes foundsome interesting area properties of parabolas.Consider the region bounded by a parabola and a chord AB. Let P be thepoint on the parabola where the tangent is parallel to the chord AB. The linethrough P parallel to the axis of the parabola meets chord AB at a point V .Then, he showed that the area of the parabolic region is a|PV | 3/2 for someconstant a, which depends only on the parabola.Furthermore, he proved that the area of the parabolic region is 4/3 timesthe area of triangle △ABP whose base is the chord and whose third vertex isP. For the proofs of Archimedes, see Chapter 7 of [8].In this paper, we consider whether this property (and similar ones) charac-terizes parabolas. As a result, we present ﬁve conditions which are necessaryand suﬃcient for a strictly convex curve in the plane to be a parabola.Usually, a curve X in the plane R

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Dong-Soo Kim

Papers

Some characterizations of spheres and elliptic paraboloids II

Helicoidal surfaces with pointwise 1-type gauss map

Finite type ruled surfaces in lorentz-minkowski space

On the archimedean characterization of parabolas

Circular cone and its Gauss map