Abstract
In 1939 Weyl showed that the volume of spherical tubes around compact submanifolds M of Euclidean space depends solely on the induced Riemannian metric on M. Can this intrinsic nature of the tube volume be preserved for tubes with more general cross sections mathbb {D} than the round ball? Under sufficiently strong symmetry conditions on mathbb {D} the answer turns out to be yes.
Highlights
Let us be given a compact connected manifold M of dimension n embedded in Rn+m as submanifold of codimension m
Of Rn+m comes with a chosen orthonormal frame in the normal bundle N M of M in Rn+m. In turn this gives for all r ∈ M an identification of the normal space Nr M with Rm, and so for Dm a compact domain around 0 in Rm we can consider the generalized tube
The main result of this paper is that under sufficiently strong symmetry requirements on the domain Dm a similar intrinsic formula for the volume V (a) of the above generalized tube remains valid as in Weyl’s case where Dm equals the unit ball Bm
Summary
Let us be given a compact connected manifold M (possibly with a boundary) of dimension n embedded in Rn+m as submanifold of codimension m. Due to the local nature of the tube formula we can assume that the submanifold M of Rn+m comes with a chosen orthonormal frame in the normal bundle N M of M in Rn+m. In turn this gives for all r ∈ M an identification of the normal space Nr M with Rm, and so for Dm a compact domain around 0 in Rm we can consider the generalized tube. If the compact domain Dm around 0 in Rm has a symmetry group Gm inside Om(R) that is orthogonal of degree n the volume of the generalized tube of type aDm for a > 0 sufficiently small is given by. We decided to leave our paper as it was, but add Sect. 8 in order to briefly survey their approach and compare their results with ours
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