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Non-existence of certain lightlike hypersurfaces of an indefinite Sasakian manifold

Samuel Ssekajja

Abstract

Here, we consider a lightlike hypersurface, tangent to the structure vector field, of an indefinite Sasakian manifold. We prove that no such a hypersurface can either have parallel or recurrent second fundamental forms. In addition to the above, we also prove that no such a hypersurface may have parallel or recurrent induced structural tensors.

Non-existence of certain lightlike hypersurfaces of an indefinite Sasakian manifold

Abstract

Here, we consider a lightlike hypersurface, tangent to the structure vector field, of an indefinite Sasakian manifold. We prove that no such a hypersurface can either have parallel or recurrent second fundamental forms. In addition to the above, we also prove that no such a hypersurface may have parallel or recurrent induced structural tensors.
Paper Structure (6 sections, 15 theorems, 85 equations)

This paper contains 6 sections, 15 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lightlike hypersurfaces of indefinite Sasakian manifolds
  4. Parallelism of second fundamental forms $h$ and $B$
  5. Recurrence of second fundamental forms $h$ and $B$
  6. Parallelism of the induced structures $\phi$, $U$ and $V$

Key Result

Theorem 2.1

Let $(M,g)$ be a lightlike hypersurface of $(\bar{M},\bar{g})$. Then, there exists a unique vector bundle $tr(TM)$, called the lightlike transversal bundle of $M$ with respect to $S(TM)$, of rank 1 over $M$ such that for any non-zero section $\xi$ of $TM^{\perp}$ on a coordinate neighbourhood $\math for any $Z$ tangent to $S(TM)$.

Theorems & Definitions (34)

  • Theorem 2.1: Duggal-Bejancu Duggal5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 24 more