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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$G$-deformations and some generalizations of H. Weyl’s tube theorem

Authors: Oldřich Kowalski and Lieven Vanhecke
Journal: Trans. Amer. Math. Soc. 294 (1986), 799-811
MSC: Primary 53C20
MathSciNet review: 825738
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Abstract: We prove an invariance theorem for the volumes of tubes about submanifolds in arbitrary analytic Riemannian manifolds under $G$-deformations of the second order. For locally symmetric spaces or two-point homogeneous spaces we give stronger invariance theorems using only $G$-deformations of the first order. All these results can be viewed as generalizations of the result of H. Weyl about isometric deformations and the volumes of tubes in spaces of constant curvature. They are derived from a new formula for the volume of a tube about a submanifold.

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Keywords: Volumes of tubes about submanifolds, Jacobi vector fields, <IMG WIDTH="22" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$G$">-deformations of order <IMG WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$k$">, locally <IMG WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="$k$">-equivalent submanifolds, total mean curvatures of tubes
Article copyright: © Copyright 1986 American Mathematical Society