The relative nullity of complex submanifolds and the Gauss map (original) (raw)
Related papers
On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms
Archiv der Mathematik, 2000
The famous Nash embedding theorem published in 1956 was aiming for the opportunity to use extrinsic help in the study of (intrinsic) Riemannian geometry, if Riemannian manifolds could be regarded as Riemannian submanifolds. However, this hope had not been materialized yet according to [23]. The main reason for this was the lack of control of the extrinsic properties of the submanifolds by the known intrinsic invariants. In order to overcome such difficulties as well as to provide answers to an open question on minimal immersions, the first author introduced in the early 1990's new types of Riemannian invariants, his so-called δ-curvatures, different in nature from the "classical" Ricci and scalar curvatures. One purpose of this article is to present some old and recent results concerning δ-invariants for Lagrangian submanifolds of complex space forms. Another purpose is to point out that the proof of Theorem 4.1 of [17] is not correct and the Theorem has to be reformulated. More precisely, Theorem 4.1 of [17] shall be replaced by Theorems 8.1 and 8.3 of this article. Since the new formulation needs a new proof, we also provide the proofs of Theorems 8.1 and 8.3 in this article.
Complex Differential Geometry: An Introduction
Let M be a 2m-dimensional topological manifold. A coordinate atlas {(U, φ U : U → C m )} is called holomorphic if the transition functions φ U • φ −1 V are holomorphic functions between subsets of C m ; in this case the coordinate charts φ U are called local holomorphic coordinates. The manifold M is called complex if it admits a holomorphic atlas. Two holomorphic atlases are called equivalent if their union is a holomorphic atlas. An equivalence class of holomorphic atlases on M is called a complex structure.
On Gauss–Bonnet and Poincaré–Hopf Type Theorems for Complex ∂ -Manifolds
Moscow Mathematical Journal, 2021
We prove a Gauss-Bonnet and Poincaré-Hopf type theorems for complex ∂-manifoldX = X − D, where X is a complex compact manifold and D is a reduced divisor. We will consider the cases such that D has isolated singularities and also if D has a (not necessarily irreducible) decomposition D = D 1 ∪ D 2 such that D 1 , D 2 have isolated singularities and C = D 1 ∩ D 2 is a codimension 2 variety with isolated singularities. As application, we obtain a generalization for the Dimca-Papadima formula.
The Gauss map for Kählerian submanifolds of bfRspn{\bf R}\sp nbfRspn
Transactions of the American Mathematical Society, 1992
We introduce a Gauss map for Kähler submanifolds of Euclidean space and study its geometry in relation to that of the given immersion. In particular we generalize a number of results of the classical theory of minimal surfaces in Euclidean space.
2006
Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. Recently, B.-Y. Chen studied Lagrangian submanifolds in complex space forms which are ideal. He proved that such submanifolds are minimal. He also classified ideal Lagrangian submanifolds in complex space forms. In the present paper, we investigate ideal Kaehlerian slant submanifolds in a complex space form. We prove that such submanifolds are minimal. We also obtain obstructions to ideal slant immersions in complex hyperbolic space and complex projective space. 2000 Mathematics Subject Classification: 53C40, 53C25. 1. Chen invariants and Chen inequalities One of the most fundamental problems in submanifold theory is the immersibility of a Riemannian manifold in a Euclidean space (or, more generally, in a space form). According to the well-known embedding theorem of Nash,...