Rigidity of time-flat surfaces in the Minkowski spacetime

Ye-Kai Wang (Department of Mathematics,Columbia University, New York, N.Y., U.S.A.; and Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Abstract

A time-flat condition on spacelike $2$-surfaces in spacetime is considered here. This condition is analogous to the constant torsion condition for curves in a three-dimensional space and has been studied in [2, 5, 6, 13, 14]. In particular, any $2$-surface in a static slice of a static spacetime is time-flat. In this paper, we address the question in the title and prove several local and global rigidity theorems for such surfaces in the Minkowski and Schwarzschild spacetimes. Higher-dimensional generalizations are also considered.

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Accepted 14 August 2014

Published 2 April 2015