Let (M,J,g) be a Hermitian manifold with complex structure J, metric g, and Kahler form Ω. Then g is locally conformal Kahler iff dΩ=ω ∧ Ω for some closed and non-exact 1-form ω. Moreover, if ω is a parallel form, M is called a generalized Hopf manifold. The main results of this paper are: (a) the description of the geometric structure of the compact locally conformal Kahler-flat manifolds; (b) the description of the geometric structure of the compact generalized Hopf manifolds on which a certain canonically defined foliation is regular; (c) a description of the harmonic forms and Betti numbers of a general compact generalized Hopf manifold; (d) a method for studying analytic vector fields on generalized Hopf manifolds; (e) conditions for submanifolds of generalized Hopf manifolds to belong to the same class.

https://link.springer.com/content/pdf/10.1007/BF00148231.pdf

Generalized Hopf manifolds

In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple $\ensuremath{\mathbb{L}^2}$ metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods.

/pdf/elastic-shape-matching-of-parameterized-surfaces-using-4t3xb0bucn.pdf

Elastic shape matching of parameterized surfaces using square root normal fields

/pdf/real-kaehler-submanifolds-and-uniqueness-of-the-gauss-map-4o20e3lzte.pdf

Real Kaehler submanifolds and uniqueness of the Gauss map

In this survey, we give an overview of Brieskorn manifolds and varieties, and their role in contact topology. We discuss open books, fillings and invariants such as contact and symplectic homology. We also present some new results involving exotic contact structures, invariants and orderability. The main tool for the required computations is a version of the Morse-Bott spectral sequence. We provide a proof for the particular version that is useful for us.

Brieskorn manifolds in contact topology

Isometric immersions with the same Gauss map

This paper investigates the relative nullity distribution of an indefi- nite Riemannian manifold isometrically immersed into an indefinite space form. Introduction. In this paper we investigate the relative nullity distribu- tion of an indefinite Riemannian manifold isometrically immersed into an indefinite space form. This distribution is totally geodesic and gives rise to a Ricatti-type differential equation along a geodesic in a leaf of the distribution. This differential equation is applied in several ways to estimate the index of relative nullity v for geodesically complete, connected, Lorentzian submanifolds M" of Af"+1(c), the Lorentzian sphere. These applications extend the work of Abe (1), (2), (3), Ferus (7), (8), and others to the setting of indefinite manifolds. Some of the work in §2 was obtained previously by Graves (10) in the codimension one case and by M. Dajczer. In particular Theorem 2 was conjectured by Dajczer (5). Sections 1 and 2 lay the groundwork and derive the Ricatti-type differential equation. In §3 an integer vn is defined and it is shown that if M" is as above and if v > vn then M" is totally geodesic. This integer is used to formulate a geometric condition which guarantees that a complete connected hypersurface of S"(c) is totally geodesic. We also estimate v given a natural condition on the space-like Ricci curvature of the sub- manifold. In (6) other conditions on Ricci curvature are given. The general scheme of our investigation is very similar to that of the Riemannian case as formulated in the papers mentioned above. However, there are a few basic and non-trivial differences from the Riemannian case. These differences are due to the indefinite metric and are to be overcome. Therefore, we think it worthwhile to include the details of the proofs for most of our results.

/pdf/relative-nullity-foliations-and-indefinite-isometric-dfzop5o9u2.pdf

Relative nullity foliations and indefinite isometric immersions.

On a Class of Hermitian Manifolds.

/pdf/on-a-generalization-of-the-hopf-fibration-i-contact-4w06qh822o.pdf

On a generalization of the Hopf fibration, I. Contact structures on the generalized Brieskorn manifolds

/pdf/nonregular-contact-structures-on-brieskorn-manifolds-47i85fovj1.pdf

Kinetsu Abe

Papers

Isometric immersions with the same Gauss map

Relative nullity foliations and indefinite isometric immersions.

On a Class of Hermitian Manifolds.

On a generalization of the Hopf fibration, I. Contact structures on the generalized Brieskorn manifolds

Nonregular contact structures on Brieskorn manifolds