Submanifolds in Normal Complex Contact Manifolds

Adela Mihai,Ion Mihai

doi:10.3390/math7121195

Abstract

In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant ( C C -totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen first inequality and Chen inequality for the invariant δ ( 2 , 2 ) for C C -totally real submanifolds in a normal complex contact space form and characterize the equality cases. We also prove the minimality of C C -totally real submanifolds of maximum dimension satisfying the equalities.

Highlights

Complex and contact geometries represent some of the most studied areas in differential geometry.The complex contact metric structures are less explored and there is a short list of papers in the mathematical literature on this topic
Blair and the first author in this work, proved that a locally symmetric normal complex contact metric manifold is locally isometric to the complex projective space C P2n+1 (4) of constant holomorphic curvature 4
If the complex contact structure is given by a global holomorphic contact form, the manifold fibers over a locally symmetric complex symplectic manifold

Summary

Introduction

Complex and contact geometries represent some of the most studied areas in differential geometry. In [2], D.E. Blair and the first author in this work, proved that a locally symmetric normal complex contact metric manifold is locally isometric to the complex projective space C P2n+1 (4) of constant holomorphic curvature 4. Blair and the first author in this work, proved that a locally symmetric normal complex contact metric manifold is locally isometric to the complex projective space C P2n+1 (4) of constant holomorphic curvature 4 They studied reflections in the integral submanifolds of the vertical subbundle of a normal complex contact manifold and showed that when such reflections are isometries the manifold fibers over a locally symmetric space. We define invariant and anti-invariant (CC-totally real) submanifolds of normal complex contact metric manifolds and start the study of their basic properties.

Preliminaries

Submanifolds

Chen Inequalities