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.)

We review part of the classical theory of curves and surfaces in 3-dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.

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Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space

First we define the notion of k-Ricci curvature of a Riemannian n- manifold. Then we establish sharp relations between the k-Ricci curvature and the shape operator and also between the k-Ricci curvature and the squared mean cur- vature for a submanifold in a Riemannian space form with arbitrary codimension. Several applications of such relationships are also presented.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C2A2DF3383777584AE7F103A29356C99/S0017089599970271a.pdf/div-class-title-relations-between-ricci-curvature-and-shape-operator-for-submanifolds-with-arbitrary-codimensions-div.pdf

Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions

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

We introduce the notion of Gauss map of pointwise 1-type on ruled surfaces in the Euclidean 3-space for which vector valued functions is neither trivial nor it extends or coincides with the usual notion of 1-type, in general. We characterize the minimal helicoid in terms of it and give a complete classification of the ruled surfaces with pointwise 1-type Gauss map.

Characterization of the helicoid as ruled surfaces with pointwise 1-type gauss map

We obtain a pointwise inequality valid for all submanifolds $M^n$ of all real space forms $N^{n+2}(c)$ with $n\ge 2$ and with codimension two, relating its main scalar invariants, namely, its scalar curvature from the intrinsic geometry of $M^n$, and its squared mean curvature and its scalar normal curvature from the extrinsic geometry of $M^n$ in $N^m(c)$.

A pointwise inequality in submanifold theory

We characterize all ruled surfaces in Minkowski 3-space with a relation between the Gauss and mean curvature (Weingarten surfaces). It turns out that, except if the rulings are in a null direction, these are given by Lorentzian screw motions of straight lines. However, if the rulings are always in a null direction, then every ruled surface is Weingarten.

Ruled Weingarten surfaces in Minkowski 3-space

We show that a ruled surface of finite type in a Euclidean space is a cylinder on a curve of finite type or a helicoid in Euclidean 3-space.

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Ruled surfaces of finite type

In an earlier paper (3) the flrst author proved that, for any isomet- ric immersion of a warped product N1 £f N2 into a Riemannian m-manifold of constant sectional curvature c, the warping function f satisfles the optimal general inequality: ¢f f • (n1 + n2)2 4n2 H 2 + n1c; where ni = dimNi, i = 1;2, H 2 is the squared mean curvature, and ¢ is the Laplacian operator of N1. Moreover, he proved in (2) that for a CR-warped product NT £f N? in a Kaehler manifold, the second fundamental form h satisfles jjhjj 2 ‚ 2n2jjr(lnf)jj 2 , where n2 = dimN?. In this article we extend these inequalities to multiply warped product manifolds in an arbitrary Rie- mannian or Kaehlerian manifold. We also provide some examples to illustrate that our results are sharp. Moreover, several applications are also obtained.

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Optimal inequalities for multiply warped product submanifolds

In a recent paper the first author introduced two sequences of Riemannian invariants on a Riemannian manifold M, denoted respectively by 6(nl,... ,nk) and 5(ni,... ,nk), which trivially satisfy 6(ni,... ,nk) > 6(n,... , nk). In this article, we completely determine the Riemannian manifolds satisfying the condition 6(nfl,... , nk) = S(nl,... ,nk). By applying the notions of these 6-invariants, we establish new characterizations of Einstein and conformally flat spaces; thus generalizing two well-known results of Singer-Thorpe and of Kulkarni.

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Franki Dillen

Papers

A pointwise inequality in submanifold theory

Ruled Weingarten surfaces in Minkowski 3-space

Ruled surfaces of finite type

Optimal inequalities for multiply warped product submanifolds

Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces