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A new integral formula for the angle between intersected submanifolds
Journal of Inequalities and Applicationsvolume 2016, Article number: 83 (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.
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].
Now we state the aim of kinematic formulas in general homogeneous spaces. Let G be a unimodular Lie group with kinematic density dg and H a closed subgroup of G. Assume that there exists an invariant Riemannian metric in the homogeneous space $G/H$. Let $M^{q}$, $N^{r}$ be two compact submanifolds of dimensions q, r in $G/H$, respectively, $M^{q}$ fixed and $gN^{r}$ the image of N under a motion $g\in G$. Let $I(M^{q}\cap gN^{r})$ denote a certain invariant of $M^{q}\cap gN^{r}$, which may be a volume, a curvature integral, etc. Then the purpose of the kinematic formula related to the invariant $I(M^{q}\cap gN^{r})$ is to evaluate the following integral:
by the wellknown integral invariants of $M^{q}$ and $N^{r}$.
The integral formulas have been studied by many geometers from various viewpoints. For example, in the case that G is the group of motions in $\mathbb{R}^{n}$, $M^{q}$, and $N^{r}$ are submanifolds of $\mathbb{R}^{n}$ and
the evaluation of (1) leads to the formulas due to Poincaré, Blaschke, Santaló, and others (see [3–5] and the references therein). Let $I(M\cap gN)=\chi(M\cap gN)$ be the Euler characteristic of $M\cap N$ of domains M and N in $\mathbb{R}^{n}$, then $\int _{G}\chi(M\cap gN)\, dg$ can be expressed explicitly by the integrals of elementary symmetric functions of principal curvatures over the boundaries and the Euler characteristics of M, N. This wellknown kinematic formula in integral geometry is due to Chern [1, 2]. Next, assume that $I(M^{q}\cap gN^{r})=\mu(M^{q}\cap gN^{r})$ is one of the integral invariants from the Weyl tube formula, then (1) leads to the ChernFederer kinematic formula for submanifolds of $\mathbb{R}^{n}$ [6]. Furthermore, Howard defined integral invariants induced from an invariant homogeneous polynomial of the second fundamental form of $M^{q}\cap gN^{r}$, and he achieved a general kinematic formula, where G is unimodular and acts transitively on the sets of tangent spaces to each of $M^{q}$ and $N^{r}$. Finally, he put the kinematic formulas listed above into a uniform shape. Most existed kinematic formulas are intrinsic, and only a few of them are extrinsic, for example, CS Chen’s formula. In [7], Zhou presented an extrinsic type kinematic formula for mean curvature powers which is a generalization of the formulas in [3, 8]. This is a typical work in which the moving frame method is used effectively. By means of some kinematic formulas, sufficient conditions for one domain to contain or to be contained in another domain can be obtained. See [5, 9–13] for more kinematic formulas and their applications.
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
Let $M^{q}$ and $N^{r}$ be two intersected submanifolds in $\mathbb{R}^{n}$ ($n\geq3$). Denote by $G(n)$ the group of rigid motions in $\mathbb{R}^{n}$ and Δ the angle between $M^{q}$ and $gN^{r}$. Then, for any positive integer k,
where $C(n)=\frac{O_{k+n}\cdots O_{k+q+1}O_{q1}\cdots O_{r+qn}O_{r1}\cdots O_{1}}{O_{k+r}\cdots O_{k+r+qn+1}O_{0}}$.
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.
The angle between intersected submanifolds
We first introduce the angle between two vector subspaces below, which will be useful for later purposes. Let V and W be vector subspaces of dimensions p and q, respectively. Let $v_{p+1}, \ldots, v_{n}$ be an orthonormal basis of $N(V)$ and $w_{q+1}, \ldots, w_{n}$ an orthonormal basis of $N(W)$, that is,
the normal spaces to V, W, respectively. The angle between subspaces V and W is defined by
where
If V, W are both $(n1)$dimensional then $\Delta(V, W)=\cos\theta $, where θ is the angle between the normals of V and W. It is obvious that
with
Similarly if g is an isometry of $\mathbb{R}^{n}$, then $\Delta(gV, gW)=\Delta(V, W)$.
Let $M^{q}$ and $N^{r}$ be two submanifolds of dimensions q, r in $\mathbb{R}^{n}$, respectively. We assume $M^{q}$ fixed and $N^{r}$ moving under the rigid motion g of $\mathbb{R}^{n}$ with the kinematic density dg. dg is the invariant measure of $G(n)$ and has the decomposition $dg= dx\, d\gamma$, where dx is the Lebesgue measure of $\mathbb{R}^{n}$ and dγ is the invariant measure of $SO(n)$. Consider generic positions $gN^{r}$ so that the intersection $M^{q}\cap g N^{r}$ is a $(q+rn)$dimensional manifold. We make use of the following convention on the ranges of indices:
Let $\{x; e_{A}\}$ be a local orthonormal frame at $x\in M^{q}$, and $e_{1}, \ldots, e_{q}$ are tangent to $M^{q}$ at x. Similarly, let $\{ x^{\prime}; e_{A}^{\prime}\}$ be a local orthonormal frame at $x^{\prime }\in gN^{r}$, and $e_{1}^{\prime}, \ldots, e_{r}^{\prime}$ are tangent to $gN^{r}$ at $x^{\prime}$. Suppose that g is generic, so that $M^{q}\cap gN^{r}$ is of dimension $q+rn$. We restrict the above families of frames by the condition
Geometrically the latter means that $x\in M^{q}\cap gN^{r}$ and $e_{\alpha}$’s are tangent to $M^{q}\cap gN^{r}$ at x. The two submanifolds $M^{q}$ and $N^{r}$ at x have a scalar invariant, the angle between $M^{q}$ and $N^{r}$, i.e.,
In the case that $M^{q}$ and $N^{r}$ are both hypersurfaces ($q=r=n1$) it is the absolute value of the cosine of the angle between their normal vectors.
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:
Moving frame 1:

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

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

$\operatorname{span} \{e_{r+1}, \ldots, e_{n}\}$ are arbitrary unit vector that complete orthonormal frame 1.
Moving frame 2:

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

$e_{r+qn+1}^{\prime}, \ldots, e_{r}^{\prime}$ 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 $e_{1}, e_{2}, \ldots, e_{r+qn}$.
By these notations, we have
with the following ranges of indices, which will be used throughout the rest of this section:
The total measure of the unoriented rplanes of $\mathbb{R}^{n}$ through a fixed point (i.e., the volume of the Grassmann manifold $G_{r, nr}$) is
where $O_{i}$ is the surface area of the idimensional unit sphere.
We also have the following density formulas:
Put
Since $e_{h}'$’s are constant vectors, we have $(e_{h}^{\prime}, de_{i})=(e_{i}, de_{h}^{\prime})=0$, and thus
From (4) and (6) we have the following formula (see [3]):
where $\Delta=\mathbf{det}(e_{r+\alpha}', e_{k}')$.
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.
Since $x\in M^{q}$, we have
where $\alpha_{h}$ and $\beta_{j}$ are 1forms. Thus
and
where Δ is the $(nr)\times(nr)$ determinant
The exterior product $\bigwedge\beta_{j}$ ($j=1, 2, \ldots, nr$) is the $(nr)$dimensional volume element on $M^{q}$ in the direction of the tangent $(nr)$plane orthogonal to $M^{q}\cap N^{r}$. Therefore, denote by $d\sigma_{q+rn}(x)$ the volume element of $M^{q}\cap N^{r}$ at x, then
where $d\sigma_{q}(x)$ is the qdimensional volume element of $M^{q}$ at x.
On the other hand, the exterior product $\omega_{1}\wedge\omega_{2} \cdots\wedge\omega_{r}$ is equal to $d\sigma_{r}(x)$ of $N^{r}$ at x. Thus, multiplying (9) by $d\sigma_{r+qn}(x)\wedge\omega_{1}\wedge \omega_{2} \cdots\wedge\omega_{r}$ and taking (10) into account, we obtain
Multiplying by $dK_{[x]}=\bigwedge\omega_{jh}$ ($j< h$; $j, h=1, 2, \ldots,n$), we have (see [3])
where $d\sigma_{r+qn}(x)$ is the volume element of $M^{q}\cap N^{r}$ at x, $d\sigma_{r}(x)$, $d\sigma_{q}(x)$ express the volume element of $M^{q}$, $N^{r}$ at x, respectively. This density formula expresses the relation of the volume element between $M^{q}$, $N^{r}$ and $M^{q}\cap gN^{r}$.
Main theorem and proof
Lemma 1
Let $L_{q[x]}$ be a fixed qplane through a fixed point x and $L_{r[x]}$ a moving rplane through x in $\mathbb{R}^{n}$. Let Δ be the angle between the two linear subspaces. Let $dL^{(2nrq)}_{nq[x]}$ denote the density of $dL_{nq[x]}$ as a subspace of the fixed $dL_{2nqr[x]}$. Assume that $r+q>n$. Then, for any positive integer k,
where $O_{i}$ is the surface area of the idimensional unit sphere.
Proof
By using (7) we have
where $dL_{r[r+qn]}$ is the density of $L_{r}$ about $L_{r+qn[x]}$ and $dL_{r+qn[x]}^{(q)}$ is the density of $L_{r+qn[x]}$ as subspace of the fixed $L_{q[x]}$. Integrating (16) over all $L_{r[x]}$, we obtain on the lefthand side the volume of the Grassmann manifold $G_{r,nr}$ and on the righthand side we can integrate $dL_{r+qn}^{(q)}$, applying the same formula (5) for $n\rightarrow q$, $r\rightarrow r+qn$, since Δ depends only on $L_{r[r+qn]}$, so we obtain
where the integral is extended over all $L_{r[r+qn]}$.
According to $dL_{r[r+qn]}=dL_{nq[x]}^{(2nrq)}$, (17) can be reformulated as
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.
Therefore (18) is rewritten as
Let $k=r+qn+a$ in (19) be a positive integer, and by (18), we arrive at
□
Lemma 2
Let $L_{r[x]}$ be the rplane through x spanned by $e_{1}, e_{2}, \ldots , e_{r}$ in $\mathbb{R}^{n}$, $dg^{r}_{[x]}$ denote the kinematic density of the group of special rotations about x in $L_{r}$, and $dg^{nr}_{[x]}$ denote the kinematic density about x in the $(nr)$plane orthogonal to $L_{r[x]}$, then
where $dg_{[x]}$ is the kinematic density of the group of special rotations about x.
Proof
By the density of linear space, we have
Let $dg^{nr}_{[x]}$ denote the kinematic density about x in $(nr)$plane orthogonal to $L_{r[x]}$, then
and
The kinematic density of the group of special rotations about x, $dg_{[x]}$ can be expressed as
From (21), (22), (23), and (24) it follows that
□
Proof of Theorem 1
By (14) and applying Lemma 2, we have
where $C(n)$ is a constant (independent of x and the manifolds $M^{q}$, $N^{r}$) that depends on the dimensions q, r and is given by the integral
taken over all positions of $N^{r}$ about x.
Next, we turn our attention to the computation of the coefficient $C(n)$. Notice that
and
where the integrals are taken over all positions of $N^{r}$ about x, so
By the density formula (16), we have
From Lemma 1, we have
Combining (26), (27), and (28), we obtain the constant in (25),
Then (25) is rewritten as
Thus we complete the proof of Theorem 1. □
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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).
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The main idea of this paper was proposed by CZ and she prepared the manuscript initially and performed all the proofs. SB and YT helped to revise the paper. All authors read and approved the final manuscript.
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MSC
 52A20
 53C65
Keywords
 kinematic formula
 integral invariant
 invariant density
 angle between two intersected submanifolds