Minimal submanifolds in certain types of kaehler product manifold

TL;DR

This paper derives a formula for the Laplacian of the second fundamental form of minimal submanifolds in certain Kaehler product manifolds, with applications to the F-anti invariant case, advancing understanding of their geometric properties.

Contribution

It provides a new formula for the Laplacian of the second fundamental form of minimal submanifolds in Kaehler product manifolds, especially in the F-anti invariant case.

Findings

Derived a formula for the Laplacian of the second fundamental form.

Discussed applications of the formula to specific submanifold cases.

Enhanced understanding of minimal submanifold geometry in Kaehler products.

Abstract

Let $M$ be a real $l$-dimensional minimal submanifold with flat normal connection in a kaehler product manifold $\overline{M}^m\times \overline{M}^n$ where $\overline{M}^m$ and $\overline{M}^n$ are complex $m$-dimensional and complex $n$-dimensional kaehler manifolds with constant holomorphic sectional curvature $c_1$ and $c_2$ respectively. We give a formula for the Laplacian of the second fundamental form of $M$. Specially we discuss the F-anti invariant case. We also give some applications of this formula.