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On extremal semi-Riemannian submanifolds. (English) Zbl 0884.53047
The author establishes a formula on the second variation of the volume functional for a submanifold of a semi-Riemannian manifold. In particular, it is proved that a semi-Riemannian submanifold cannot be extremal, unless it is both definite and codefinite.
MSC:
53C42Immersions (differential geometry)
53C50Lorentz manifolds, manifolds with indefinite metrics
References:
[1]Abraham R., Marsden J.E., Tudor R.,Manifolds, Tensor analysis, and Applications, Addison-Wesley, 1983 Reading.
[2]Brill D., Flaherty F.,Isolated maximal surfaces in spacetime. Commun. Math. Phys.,50 (1976), 157–165. · Zbl 0337.53051 · doi:10.1007/BF01617993
[3]Ishihara T.,On the Stability of Maximal Submanifolds in Pseudo–Riemannian Manifolds, J. Math. Tokushima Univ.,21 (1987), 1–11.
[4]Lawson B.H.,Lectures on Minimal Submanifolds, Publish or Perish, 1980 Berkeley.
[5]Nomizu K.,Remarks on Sectional Curvature of an Indefinite Metric, Proc. Amer. Math. Soc.,89 (1983), 473–416. · doi:10.1090/S0002-9939-1983-0715869-0
[6]O’Neill B.,Semi-Riemannian Geometry, Academic Press, 1983 New York.
[7]Poor W.A.,Differential Geometric Structures, McGraw-Hill, 1981 New York.