Abstract

The aim of this paper is to derive the first variation of a functional on an oriented submanifold in the Riemannian manifold involving an arbitrary vector field. After obtaining the first variation formula, a notion of $\sigma$-mean curvature is naturally introduced. We also find an important identity involving the $\sigma$-mean curvature and the divergence of the tangent component of the vector field. As an application, we get a general formula of the $\sigma$-mean curvature for the hypersurfaces in a class of Finsler manifolds called the general $(\alpha,\beta)$-manifolds (introduced in [38]). Hence the vanishing $\sigma$-mean curvature characterizes the minimal hypersurfaces under the Busemann-Hausdorff measure ([29]) and the Holmes-Thompson measure ([23]). We also give a general formula for the hypersurface in a Randers manifold involving the navigation data without any restriction on the vector field. In terms of the identity, we prove some nonexistence theorems of the closed orientable minimal submanifold in some special non-Minkowskian Finsler manifolds.

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