R E S E A R C H
Open Access
A new integral formula for the angle
between intersected submanifolds
Chunna Zeng
*, Shikun Bai and Yin Tong
*Correspondence:
College of Mathematics Science, Chongqing Normal University, Chongqing, 401331, People’s Republic of China
Abstract
LetGbe a Lie group andHits subgroup, andMq,Nrtwo submanifolds of dimensions q,r, respectively, in the Riemannian homogeneous spaceG/H. A kinematic integral formula for the angle between the two intersected submanifolds is obtained.
MSC: 52A20; 53C65
Keywords: kinematic formula; integral invariant; invariant density; angle between two intersected submanifolds
1 Introduction
Kinematic formulas in integral geometry are important and useful. At the beginning of [], 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 inte-gral invariants’. He proved the fundamental kinematic formula inn-dimensional Euclidean spaceRnin []. In [], 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 [] and [].
Now we state the aim of kinematic formulas in general homogeneous spaces. LetGbe a unimodular Lie group with kinematic densitydgandHa closed subgroup ofG. Assume that there exists an invariant Riemannian metric in the homogeneous spaceG/H. LetMq,
Nrbe two compact submanifolds of dimensionsq,rinG/H, respectively,Mqfixed and
gNr the image ofN under a motion g∈G. LetI(Mq∩gNr) denote a certain invariant
ofMq∩gNr, which may be a volume, a curvature integral,etc.Then the purpose of the
kinematic formula related to the invariantI(Mq∩gNr) is to evaluate the following integral:
{g∈G:Mq∩gNr=∅}I
Mq∩gNrdg ()
by the well-known integral invariants ofMqandNr.
The integral formulas have been studied by many geometers from various viewpoints. For example, in the case thatGis the group of motions inRn,Mq, andNrare submanifolds
ofRnand
IMq∩gNr=volMq∩gNr, ()
the evaluation of () leads to the formulas due to Poincaré, Blaschke, Santaló, and others (see [–] and the references therein). LetI(M∩gN) =χ(M∩gN) be the Euler charac-teristic of M∩N of domainsMandN inRn, then
Gχ(M∩gN)dg can be expressed
explicitly by the integrals of elementary symmetric functions of principal curvatures over the boundaries and the Euler characteristics ofM,N. This well-known kinematic formula in integral geometry is due to Chern [, ]. Next, assume thatI(Mq∩gNr) =μ(Mq∩gNr)
is one of the integral invariants from the Weyl tube formula, then () leads to the Chern-Federer kinematic formula for submanifolds ofRn[]. Furthermore, Howard defined
in-tegral invariants induced from an invariant homogeneous polynomial of the second fun-damental form ofMq∩gNr, and he achieved a general kinematic formula, whereGis
unimodular and acts transitively on the sets of tangent spaces to each ofMqandNr. Fi-nally, he put the kinematic formulas listed above into a uniform shape. Most existed kine-matic formulas are intrinsic, and only a few of them are extrinsic, for example, C-S Chen’s formula. In [], Zhou presented an extrinsic type kinematic formula for mean curvature powers which is a generalization of the formulas in [, ]. This is a typical work in which the moving frame method is used effectively. By means of some kinematic formulas, suf-ficient conditions for one domain to contain or to be contained in another domain can be obtained. See [, –] for more kinematic formulas and their applications.
In addition, Zhou stated in [] that an important unsolved problem is whether an in-variantI(Mq∩gNr) (either intrinsic or extrinsic) can be expressed by invariants of sub-manifoldsMqandNq. At least we are not aware of lettingI(M∩gN) =diam(M∩gN), the
diameter of intersectionI(M∩gN) of two domainsMandNinRn. LetI(Mq∩gNr) be the
angle between two intersected submanifolds (the angle betweenMqandgNris an integral
invariant []) and the explicit formula is still obscure to us. In this paper, we will discuss the second problem and obtain an extrinsic kinematic formula.
LetG(n) be the group of rigid motions inRnandO(n) the group of rotations inRn.
Denote bydgthe invariant measure of the groupG(n) which is the product measure of the Lebesgue measure ofRnand the invariant measure ofSO(n), where the invariant measure
of SO(n) is normalized so that the total measure is On–· · ·O. LetMq andNr be two submanifolds inRn,Mq fixed, andgNr moving under the rigid motiongofRnwith the
kinematic densitydg. Denote bydσthe volume element.
Theorem Let Mqand Nrbe two intersected submanifolds inRn(n≥).Denote by G(n)
the group of rigid motions inRnandthe angle between Mqand gNr.Then,for any positive
integer k,
G(n)
Mq∩gNr
kdσr+q–ndg=C(n)σq
Mqσr
Nr, ()
where C(n) =Ok+n···Ok+q+Oq–···Or+q–nOr–···O
Ok+r···Ok+r+q–n+O .
2 Preliminaries
In this section, we review some basic facts about the angle between intersected submani-folds inRnand present an important density formula.
2.1 The angle between intersected submanifolds
Letvp+, . . . ,vnbe an orthonormal basis ofN(V) andwq+, . . . ,wnan orthonormal basis of
N(W), that is,
N(V) = span{vp+, . . . ,vn};
N(W) = span{wq+, . . . ,wn},
the normal spaces toV,W, respectively. The angle between subspacesVandWis defined by
(V,W) =vp+∧ · · · ∧vn∧wq+∧ · · · ∧wn,
where
x∧ · · · ∧xk=det
(xi,xs), ≤i,s≤k.
IfV,Ware both (n– )-dimensional then(V,W) =|cosθ|, whereθis the angle between the normals ofVandW. It is obvious that
≤(V,W)≤,
with
(V,W) = if and only if V∩W={},
(V,W) = if and only if V⊥W.
Similarly ifgis an isometry ofRn, then(gV,gW) =(V,W).
LetMqandNrbe two submanifolds of dimensionsq,rinRn, respectively. We assume
Mq fixed andNrmoving under the rigid motiongofRnwith the kinematic densitydg.
dg is the invariant measure ofG(n) and has the decompositiondg=dx dγ, wheredxis the Lebesgue measure ofRnanddγ is the invariant measure ofSO(n). Consider generic
positionsgNrso that the intersectionMq∩gNris a (q+r–n)-dimensional manifold. We
make use of the following convention on the ranges of indices:
≤A≤n; ≤α≤q+r–n; q+r–n+ ≤a≤q;
q+ ≤λ≤n; q+r–n+ ≤h≤r; r+ ≤ρ≤n.
Let{x;eA}be a local orthonormal frame atx∈Mq, ande, . . . ,eqare tangent toMqatx.
Similarly, let{x;eA}be a local orthonormal frame atx∈gNr, ande
, . . . ,erare tangent to
gNratx. Suppose thatgis generic, so thatMq∩gNris of dimensionq+r–n. We restrict
the above families of frames by the condition
x=x, eα=eα.
Nr,i.e.,
=deteλ,eρ=det
ea,eh.
In the case thatMqandNrare both hypersurfaces (q=r=n– ) it is the absolute value of the cosine of the angle between their normal vectors.
2.2 Some relations between densities of linear subspaces
LetObe a fixed point (origin) and letLq[O] be a fixedq-plane throughO. LetLr[O]be a movingr-plane throughOand assume thatq+r>n, so thatLq[O]∩Lr[O] is, in general, a (r+q–n)-plane throughO, which we represent byLr+q–n[O]. We can expressdLr[O]as a product ofdLr[r+q–n](density ofLraboutLr+q–n[O]) anddL(rq+)q–n[O](density ofLr+q–n[O]as subspace of the fixed aboutLq[O]). We consider the following two orthonormal moving frames:
Moving frame:
• span{e,e, . . . ,er+q–n}=Lq[O]∩Lr[O];
• er+q–n+, . . . ,erlie onLr[O];
• span{er+, . . . ,en}are arbitrary unit vector that complete orthonormal frame .
Moving frame:
• span{e,e, . . . ,er+q–n}=Lq[O]∩Lr[O];
• er+q–n+, . . . ,erare constant unit vectors in the(n–q)-planeLn–q[O]perpendicular to
Lq[O];
• eq+, . . . ,enare contained inLq[O]such that they form an orthonormal frame inLq[O]
together withe,e, . . . ,er+q–n.
By these notations, we have
dLr[O]=
α,i
(er+α,dei)
α,h
(er+α,deh) ()
with the following ranges of indices, which will be used throughout the rest of this section:
α= , , . . . ,n–q; i= , , . . . ,r+q–n;
h=r+q–n+ , . . . ,q; k=r+ ,r+ , . . . ,n.
The total measure of the unorientedr-planes ofRnthrough a fixed point (i.e., the volume
of the Grassmann manifoldGr,n–r) is
m(Gr,n–r) =m(Gn–r,r) =
Gr,n–r
dLr[O]=
On–On–· · ·On–r
Or–Or–· · ·OO
, ()
whereOiis the surface area of thei-dimensional unit sphere.
We also have the following density formulas:
dLr[r+q–n]=
α,h
(er+α,deh),
dL(rq+)q–n[O]=
α,i
er+α,dei
Put
er+α=
h
ur+αeh+
k
ur+α,kek.
Sinceeh’s are constant vectors, we have (eh,dei) = –(ei,deh) = , and thus
(er+α,dei) =
k
ur+α,k
ek,dei
. ()
From () and () we have the following formula (see []):
dLr[O]=r+q–ndLr[r+q–n]∧dLr(q+)q–n[O], ()
where= det(er+α,ek).
2.3 An important differential formula
LetMq be a fixedq-dimensional manifold andNr a moving one of dimensionr, both
assumed smooth of classC, having finite volumesσ
q(Mq) andσr(Nr), respectively. Let
q+r≥nand consider positions ofNrsuch thatMq∩Nr=φ. Letx∈Mq∩Nrand choose the orthonormal vectors e,e, . . . ,en such that e,e, . . . ,er+q–n are tangent to Mq∩Nr
and er+q–n+, . . . ,er are tangent to Nr. Lete, . . . ,en–r be orthonormal vectors such that
e,e, . . . ,er+q–n,e, . . . ,en–rspan the tangentq-plane toMqatx.
Sincex∈Mq, we have
dx=
r+q–n
h=
αheh+ n–r
j=
βjej, ()
whereαhandβjare -forms. Thus
ωr+h=dx·er+h= n–r
j=
βj
ej,er+h
, h= , , . . . ,n–r ()
and
n–r
h=
ωr+h= n–r
j=
βj, ()
whereis the (n–r)×(n–r) determinant
=ej,er+h (j,h= , , . . . ,n–r). ()
The exterior product βj(j= , , . . . ,n–r) is the (n–r)-dimensional volume element
onMq in the direction of the tangent (n–r)-plane orthogonal toMq∩Nr. Therefore,
denote bydσq+r–n(x) the volume element ofMq∩Nratx, then
dσr+q–n(x) n–r
j=
βj=dσq(x), ()
On the other hand, the exterior productω∧ω· · · ∧ωr is equal todσr(x) ofNr atx.
Thus, multiplying () by dσr+q–n(x)∧ω∧ω· · · ∧ωr and taking () into account, we
obtain
dσr+q–n(x) n
i=
ωi=dσq(x)∧dσr(x). ()
Multiplying bydK[x]= ωjh(j<h;j,h= , , . . . ,n), we have (see [])
dσr+q–n(x)∧dK=dσq(x)∧dσr(x)∧dK[x], ()
wheredσr+q–n(x) is the volume element ofMq∩Nratx,dσr(x),dσq(x) express the volume
element ofMq,Nr atx, respectively. This density formula expresses the relation of the
volume element betweenMq,NrandMq∩gNr.
3 Main theorem and proof
Lemma Let Lq[x]be a fixed q-plane through a fixed point x and Lr[x] a moving r-plane
through x inRn.Letbe the angle between the two linear subspaces.Let dL(n–r–q) n–q[x] denote
the density of dLn–q[x]as a subspace of the fixed dLn–q–r[x].Assume that r+q>n.Then,for
any positive integer k,
Gn–q,n–r
kdL(n–nq–[rx–]q)=Ok+n–q–r–· · ·Ok+n–r
Ok+n–q–· · ·Ok
, ()
where Oiis the surface area of the i-dimensional unit sphere.
Proof By using () we have
dLr[x]=r+q–ndLr[r+q–n]∧dLr(q+)q–n[x], ()
wheredLr[r+q–n]is the density ofLraboutLr+q–n[x]anddL(rq+)q–n[x]is the density ofLr+q–n[x] as subspace of the fixedLq[x]. Integrating () over allLr[x], we obtain on the left-hand side the volume of the Grassmann manifoldGr,n–rand on the right-hand side we can integrate
dL(rq+)q–n, applying the same formula () forn→q,r→r+q–n, sincedepends only on
Lr[r+q–n], so we obtain
Gn–q,n–r
r+q–ndLr[r+q–n]=
On–On–· · ·Oq
Or–Or–· · ·Or+q–n
, ()
where the integral is extended over allLr[r+q–n].
According todLr[r+q–n]=dLn(–nq–[rx–]q), () can be reformulated as
Gn–q,n–r
r+q–ndLnn––qr[x–]q= On–· · ·Oq
Or–· · ·Or+q–n
. ()
Supposea∈[n–r–q+ , +∞] is an integer and make the change of notationn→n+a,
Therefore () is rewritten as
Gn–q,n–r
r+q–n+adLn(–nq–[rx–]q)= On+a–· · ·Oq+a
Or+a–· · ·Or+q–n+a
. ()
Letk=r+q–n+ain () be a positive integer, and by (), we arrive at
Gn–q,n–r
kdL(n–nq–[rx–]q)=Ok+n–r–q–· · ·Ok+n–r
Ok+n–q–· · ·Ok
. ()
Lemma Let Lr[x] be the r-plane through x spanned by e,e, . . . ,er inRn,dg[rx] denote
the kinematic density of the group of special rotations about x in Lr,and dg[nx–]rdenote the
kinematic density about x in the(n–r)-plane orthogonal to Lr[x],then
dg[x]=dg[rx]∧dLr[x]∧dg[nx–]r,
where dg[x]is the kinematic density of the group of special rotations about x.
Proof By the density of linear space, we have
dLr[x]=
α,β
ωαβ, α= , , . . . ,r;β=r+ , . . . ,n. ()
Letdg[nx–]rdenote the kinematic density aboutxin (n–r)-plane orthogonal toLr[x], then
dg[nx–]r=
λ<μ
ωλμ, λ,μ=r+ , . . . ,n ()
and
dg[rx]=
c<f
ωcf, c,f = , . . . ,r. ()
The kinematic density of the group of special rotations aboutx,dg[x]can be expressed as
dg[x]=
B<C
ωBC, B,C= , . . . ,n. ()
From (), (), (), and () it follows that
dg[x]=dg[rx]∧dLr[x]∧dg[nx–]r.
Proof of Theorem By () and applying Lemma , we have
G(n)
Mq∩gNr
kdσr+q–n(x)∧dg=
G(n)
Mq∩gNr
k+dσq(x)∧dσr(x)∧dg[x]
=C(n)σq
Mqσr
whereC(n) is a constant (independent ofxand the manifoldsMq,Nr) that depends on
the dimensionsq,rand is given by the integral
C(n) =
k+dg[x]
taken over all positions ofNraboutx.
Next, we turn our attention to the computation of the coefficientC(n). Notice that
C(n) =
k+dg[x]=
k+dg[rx]∧dLr[x]∧dg[nx–]r
and
dgr[x]=Or–· · ·O,
dg[nx–]r=On–r–· · ·O,
where the integrals are taken over all positions ofNraboutx, so
k+dg[x]=Or–· · ·OOn–r–· · ·O
k+dLr[x]. ()
By the density formula (), we have
k+dLr[x]=
k+r+q–ndLr[r+q–n]∧dL(rq+)q–n[x]
= Oq–· · ·On–r
Or+q–n–· · ·OO
k+r+q–n+dLn(–nq–[rx–]q). ()
From Lemma , we have
k+r+q–n+dLn(–nq–[rx–]q)= Ok+n· · ·Ok+q+
Ok+r· · ·Ok+r+q–n+
. ()
Combining (), (), and (), we obtain the constant in (),
C(n) = Ok+n· · ·Ok+q+Oq–· · ·Or+q–nOr–· · ·O
Ok+r· · ·Ok+r+q–n+O
.
Then () is rewritten as
G(n)
Mq∩gNr
kdσr+q–n∧dg
=σq
Mqσr
NrOk+n· · ·Ok+q+Oq–· · ·Or+q–nOr–· · ·O Ok+r· · ·Ok+r+q–n+O
.
Thus we complete the proof of Theorem .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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).
Received: 14 September 2015 Accepted: 19 February 2016
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