FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 1, PAGES 245-262

On an application of the Stokes' theorem in global Riemannian geometry

S. E. Stepanov

Abstract

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Applying the Stokes' theorem we have deduced the Weitzenbock's formula for symmetric $2$-forms on a compact Riemannian manifold $M$ with boundary $$¶M ¹ Æ. Using the formula we have proved that Killing symmetric $2$-forms and Killing $p$-forms on a Riemannian manifold $M$ of non-positive sectional curvature and convex boundary $$¶M must be either parallel or zero. Finally, we have applied our results to the global theory of projective and umbilical maps.

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Last modified: July 8, 2002