A new integral formula for the angle between intersected submanifolds
 Chunna Zeng^{1}Email author,
 Shikun Bai^{1} and
 Yin Tong^{1}
DOI: 10.1186/s136600161028x
© Zeng et al. 2016
Received: 14 September 2015
Accepted: 19 February 2016
Published: 2 March 2016
Abstract
Let G be a Lie group and H its subgroup, and \(M^{q}\), \(N^{r}\) two submanifolds of dimensions q, r, respectively, in the Riemannian homogeneous space \(G/H\). A kinematic integral formula for the angle between the two intersected submanifolds is obtained.
Keywords
kinematic formula integral invariant invariant density angle between two intersected submanifoldsMSC
52A20 53C651 Introduction
Kinematic formulas in integral geometry are important and useful. At the beginning of [1], Chern said: ‘one of the basic problems in integral geometry is to find explicit formulas for the integrals of geometric quantities over the kinematic density in terms of known integral invariants’. He proved the fundamental kinematic formula in ndimensional Euclidean space \(\mathbb{R}^{n}\) in [2]. In [1], he provided integral formulas for the quantities introduced by Weyl for the volume of tubes. These formulas complement the fundamental kinematic formula, which only deals with hypersurfaces. They can be found in [3] and [4].
In addition, Zhou stated in [7] that an important unsolved problem is whether an invariant \(I(M^{q}\cap gN^{r})\) (either intrinsic or extrinsic) can be expressed by invariants of submanifolds \(M^{q}\) and \(N^{q}\). At least we are not aware of letting \(I(M\cap gN)=\operatorname{diam}( M\cap gN)\), the diameter of intersection \(I(M\cap gN)\) of two domains M and N in \(\mathbb{R}^{n}\). Let \(I(M^{q}\cap gN^{r})\) be the angle between two intersected submanifolds (the angle between \(M^{q}\) and \(gN^{r}\) is an integral invariant [1]) and the explicit formula is still obscure to us. In this paper, we will discuss the second problem and obtain an extrinsic kinematic formula.
Let \(G(n)\) be the group of rigid motions in \(\mathbb{R}^{n}\) and \(O(n)\) the group of rotations in \(\mathbb{R}^{n}\). Denote by dg the invariant measure of the group \(G(n)\) which is the product measure of the Lebesgue measure of \(\mathbb{R}^{n}\) and the invariant measure of \(SO(n)\), where the invariant measure of \(SO(n)\) is normalized so that the total measure is \(O_{n1}\cdots O_{1}\). Let \(M^{q}\) and \(N^{r}\) be two submanifolds in \(\mathbb{R}^{n}\), \(M^{q}\) fixed, and \(gN^{r}\) moving under the rigid motion g of \(\mathbb{R}^{n}\) with the kinematic density dg. Denote by dσ the volume element.
Theorem 1
2 Preliminaries
In this section, we review some basic facts about the angle between intersected submanifolds in \(\mathbb{R}^{n}\) and present an important density formula.
2.1 The angle between intersected submanifolds
2.2 Some relations between densities of linear subspaces
Let O be a fixed point (origin) and let \(L_{q[O]}\) be a fixed qplane through O. Let \(L_{r[O]}\) be a moving rplane through O and assume that \(q+r>n\), so that \(L_{q[O]}\cap L_{r[O]}\) is, in general, a \((r+qn)\)plane through O, which we represent by \(L_{r+qn[O]}\). We can express \(dL_{r[O]}\) as a product of \(dL_{r[r+qn]}\) (density of \(L_{r}\) about \(L_{r+qn[O]}\)) and \(dL_{r+qn[O]}^{(q)}\) (density of \(L_{r+qn[O]}\) as subspace of the fixed about \(L_{q[O]}\)). We consider the following two orthonormal moving frames:

\(\operatorname{span}\{e_{1}, e_{2}, \ldots, e_{r+qn}\}=L_{q[O]}\cap L_{r[O]}\);

\(e_{r+qn+1}, \ldots, e_{r}\) lie on \(L_{r[O]}\);

\(\operatorname{span} \{e_{r+1}, \ldots, e_{n}\}\) are arbitrary unit vector that complete orthonormal frame 1.

\(\operatorname{span}\{e_{1}, e_{2}, \ldots, e_{r+qn}\}=L_{q[O]}\cap L_{r[O]}\);

\(e_{r+qn+1}^{\prime}, \ldots, e_{r}^{\prime}\) are constant unit vectors in the \((nq)\)plane \(L_{nq}[O]\) perpendicular to \(L_{q[O]}\);

\(e_{q+1}^{\prime}, \ldots, e_{n}^{\prime}\) are contained in \(L_{q[O]}\) such that they form an orthonormal frame in \(L_{q[O]}\) together with \(e_{1}, e_{2}, \ldots, e_{r+qn}\).
2.3 An important differential formula
Let \(M^{q}\) be a fixed qdimensional manifold and \(N^{r}\) a moving one of dimension r, both assumed smooth of class \(C^{1}\), having finite volumes \(\sigma _{q}{(M^{q})}\) and \(\sigma_{r}{(N^{r})}\), respectively. Let \(q+r\geq n\) and consider positions of \(N^{r}\) such that \(M^{q}\cap N^{r}\neq\phi\). Let \(x\in M^{q}\cap N^{r}\) and choose the orthonormal vectors \(e_{1}, e_{2}, \ldots, e_{n}\) such that \(e_{1}, e_{2}, \ldots, e_{r+qn}\) are tangent to \(M^{q}\cap N^{r}\) and \(e_{r+qn+1}, \ldots, e_{r}\) are tangent to \(N^{r}\). Let \(e_{1}^{\prime}, \ldots, e_{nr}^{\prime}\) be orthonormal vectors such that \(e_{1}, e_{2}, \ldots, e_{r+qn}\), \(e_{1}^{\prime}, \ldots, e_{nr}^{\prime}\) span the tangent qplane to \(M^{q}\) at x.
3 Main theorem and proof
Lemma 1
Proof
Suppose \(a\in[nrq+1,+\infty]\) is an integer and make the change of notation \(n\rightarrow n+a\), \(r\rightarrow r+a\), \(q\rightarrow q+a\). Since the increased dimension is the dimension of the intersected part, the angle remains unchanged.
Lemma 2
Proof
Proof of Theorem 1
Thus we complete the proof of Theorem 1. □
Declarations
Acknowledgements
The authors would like to thank two anonymous referees for many helpful comments and suggestions that directly lead to the improvement of the original manuscript. The authors are supported in part by Natural Science Foundation Project of CQ CSTC (Grant No. cstc 2014jcyjA00019), NSFC (Grant No. 11326073), Technology Research Foundation of Chongqing Educational Committee (Grant No. KJ1500312), and 2014 Natural Foundation advanced research of CQNU (Grant No. 14XYY028).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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