On $(2k)$-Minimal Submanifolds

Details On $(2k)$-Minimal Submanifolds On $(2k)$-Minimal Submanifolds

2007-06-21

arxiv.org/abs/0706.3092v1

Mathematics

Differential Geometry

18 pages

Scientific paper

Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total $(2k)$-th Gauss-Bonnet curvature function, called $(2k)$-minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the $(2k+1)$-Gauss-Bonnet curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to $(2k)$-minimal submanifolds.

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