Integral Formulae

Abstract

AbstractThe chapter presents a series of new Integral Formulae (IF) for a codimension-one foliation on a closed Riemannian manifold. The proof of IF is based on the Divergence Theorem. The IF start from the formula by Reeb, for foliations on space forms they generalize the classical ones by Asimov, Brito, Langevin, and Rosenberg. Our IF include also a set of arbitrary functions f j depending on the scalar invariants of the Weingarten operator. For a special choice of auxiliary functions the IF involve the Newton transformations of the Weingarten operator. We apply IF to umbilical foliations and foliations whose leaves have constant second-order mean curvature.KeywordsVector FieldRiemannian ManifoldFundamental FormIntegral FormulaShape OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.