aa r X i v : . [ m a t h . DG ] F e b New Hsiung-Minkowski identities
Rui AlbuquerqueFebruary 18, 2021
Abstract
We find the first three most general Minkowski or Hsiung-Minkowski identities relating thetotal mean curvatures H i , of degrees i = 1 , ,
3, of a closed hypersurface N immersed in agiven orientable Riemannian manifold M endowed with any given vector field P . Then wespecialise the three identities to the case when P is a position vector field. We further obtainthat the classical Minkowski identity is natural to all Riemannian manifolds and, moreover, thata corresponding 1st degree Hsiung-Minkowski identity holds true for all Einstein manifolds. Weapply the result to hypersurfaces with constant H , H . Key Words: exterior differential system, hypersurface, i th mean curvature, Einstein metric. MSC 2020:
Primary: 53C21, 53C25, 53C65; Secondary: 53C17, 53C42, 57R25, 58A15
1. Introduction
Let M denote an orientable Riemannian manifold of class C and dimension n + 1. Let P ∈ X U be a position vector field of M defined on an open subset U ⊂ M . This is, there exists a function f ∈ C U such that, for any X ∈ X U , ∇ X P = f X. (1)Let N be a closed orientable isometrically immersed hypersurface of M of class C , contained in U . Let ν denote one of the two unit normals to N in M and let H i denote the i th mean curvatureof N with respect to ν . As usual, H = 1. Now we may recall the Hsiung-Minkowski identities: if M has constant sectional curvature, then, for 0 ≤ i ≤ n − Z N ( f H i − h P, ν i H i +1 ) vol N = 0 . (2)This identity was found by Hsiung in [7] in the case of Euclidean space, with position vector field P x = x ∈ T x M , hence with f = 1, generalizing the same result of Minkowski for i = 0.The Hsiung-Minkowski identities were further improved by Y. Katsurada, in [11], who foundcertain generalized Minkowski identities for a conformal-Killing vector field, in place of P , stillon the ambient of a constant sectional curvature M contained in Euclidean space. The true for-mula and meaning of a position vector field there remains a bit obscure. Nevertheless, since P is1 . Albuquerque i = 0 , ,
2. Indeed generalized, in as much as they are concernedwith any vector field and any oriented Riemannian manifold M .Given a position vector field P , we show in Theorem 2.2 that Z N ( f − h P, ν i H ) vol = 0 (3)and Z N (cid:16) (cid:18) n (cid:19) f H − (cid:18) n (cid:19) h P, ν i H − Ric(
P, ν ) + h P, ν i Ric( ν, ν ) (cid:17) vol = 0 (4)and Z N (cid:16)(cid:0) h P, ν i Ric( ν, ν ) − Ric( ν, P ) (cid:1) nH + 3 (cid:18) n (cid:19) f H − (cid:18) n (cid:19) h P, ν i H ++ h P, ν i tr( R ( ν, ∇ · ν ) ν ) + Ric( ∇ P ν, ν ) − tr( R ( P, ∇ · ν ) ν ) (cid:17) vol = 0 . (5)We then find that (2) is valid for every Riemannian manifold in case i = 0, valid for everyEinstein manifold in cases i = 0 ,
1, and, still as a corollary, valid for constant sectional curvaturein cases i = 0 , ,
2. Let us remark that every warped product metric admits a position vector field.We end the article with some applications to the theory of isoparametric hypersurfaces in a givenEinstein manifold.This article brings to light a new application of a fundamental differential system of Riemanniangeometry introduced in [1]. Given our self-contained methods in the treatment of the i th-meancurvatures and beyond, we again recommend the reading of [2].
2. Recalling the fundamental differential system
Our framework for the proof of the generalized integral identities is that of the unit tangentsphere bundle π : SM −→ M together with its exterior differential system of invariant n -formson the (2 n + 1)-dimensional manifold SM . Let us start by recalling such fundamental differentialsystem. We use the notation h , i for the metric.We take the tangent bundle π : T M −→ M with the canonical Sasaki metric and SO( n + 1)structure and recall the mirror map B ∈ End
T T M restricted to SM , see [1, 2]. The map ischaracterized by B ◦ B = 0 and Bπ ∗ X = π ⋆ X , for any X tangent to M . The notation is theusual: π ∗ for the horizontal lift, to the pullback tangent bundle identified via d π with horizontalsin T T M = π ∗ T M ⊕ π ⋆ T M , and π ⋆ for the vertical lift to another copy of the pullback tangentbundle, identified with verticals π ⋆ T M = ker d π only due to the differential structure. . Albuquerque ξ , that which is defined as the vertical ξ u = u ∈ T T M , ∀ u ∈ T M , yields a unit horizontal vector field e over SM and a dual 1-form θ = e . This definesa well-known contact structure. Moreover, e is the restriction to SM of the geodesic flow vectorfield. Its mirror Be = ξ is the canonical vector field on T M . Hence we have a reduction of thestructural group of the Riemannian manifold SM to SO( n ).We further recall that a frame e , e , . . . , e n , e n +1 , . . . , e n is said to be an adapted frame on SM if: it is orthonormal, the first n + 1 vectors are horizontal, the remaining are vertical and each ofthe e i + n is the mirror of e i .Now let us recall the differential system of M found in [1]. This is a collection of global n -forms α i on SM defined as follows. First we let α n = ξ y π ⋆ vol be the volume form of the fibres. Then foreach i = 0 , . . . , n we let α i = 1 i !( n − i )! α n ◦ ( B n − i ∧ i ) . (6) ◦ denotes an alternating operator which is particularly efficient under differentiation: Leibniz ruleapplies in the obvious way, with no signs attached. It is convenient to recall here, for any manifold,any 1-forms η , . . . η p and any endomorphisms B , . . . , B p , the result that( η ∧ · · · ∧ η p ) ◦ ( B ∧ · · · ∧ B p ) = X σ ∈ Sym p η ◦ B σ ∧ · · · ∧ η p ◦ B σ p , (7)so the reader may know precisely what we are referring. Notice α n = e n +1 ∧ · · · ∧ e n + n and we maysay i is the number of vertical directions in each summand of the n -form α i . For convenience onealso defines α − = α n +1 = 0.Given an orientable C immersed hypersurface ι : N ֒ → M with the induced metric, let us recallthe second fundamental form is defined by A = ∇ ν , where ν is one of the two unit-normals to N .Then, for 0 ≤ i ≤ n , the i th-mean curvature H i is defined by (cid:0) ni (cid:1) H i being the elementary symmetricpolynomial of degree i on the eigenvalues a , . . . , a n of A , the so-called principal curvatures of N .In other words, H = 1 and (cid:0) ni (cid:1) H i = P ≤ j < ··· Let ι : N ֒ → M be a closed orientable isometrically immersed C hypersurface of M . Let P be any vector field of M defined on a neighborhood of ι ( N ) . Then we have that Z N n !ˆ ι ∗ ( α n ◦ ( L P B n )) − (cid:18) n (cid:19) h P, ν i H vol = 0 (13) and Z N n − ι ∗ ( α n ◦ ( L P B n − ∧ − (cid:16) (cid:18) n (cid:19) h P, ν i H + Ric( P, ν ) − h P, ν i Ric( ν, ν ) (cid:17) vol = 0 (14) and Z N n − ι ∗ ( α n ◦ ( L P B n − ∧ )) + (cid:16)(cid:0) h P, ν i Ric( ν, ν ) − Ric( ν, P ) (cid:1) nH − (cid:18) n (cid:19) h P, ν i H + h P, ν i tr( R ( ν, ∇ · ν ) ν ) + Ric( ∇ P ν, ν ) − tr( R ( P, ∇ · ν ) ν ) (cid:17) vol = 0 . (15) Proof. Regarding L P B , notice we are denoting also by P the horizontal lift π ∗ P of P to SM .Moreover, the horizontal lift P is tangent to SM , so we can apply Lie derivative to any vectorbundle sections restricted to SM .As with ˆ ι ∗ θ = 0, it will become necessary to see, what is quite immediate, thatˆ ι ∗ ( P y θ ) = ˆ ι ∗ ( θ ( P )) = h ν, P i , ˆ ι ∗ r = Ric( ν, ν ) . (16)Now, applying the well established Leibniz rule, cf. [1], on (10) and noticing P y α n = 0, since P is horizontal, on the one hand we have L P α i = 1 i !( n − i )! (cid:16) ( L P α n ) ◦ ( B n − i ∧ i ) + α n ◦ ( L P B n − i ∧ i ) (cid:17) = 1 i !( n − i )! ( P y R ξ α n ) ◦ ( B n − i ∧ i ) + 1 i !( n − i )! α n ◦ ( L P B n − i ∧ i ) . And, by Cartan’s formula again, on the other hand we have L P α i = d( P y α i ) + P y (cid:0) ( i + 1) θ ∧ α i +1 + R ξ α i (cid:1) = d( P y α i ) + ( i + 1) h P, ν i α i +1 − ( i + 1) θ ∧ P y α i +1 + P y R ξ α i . The identities in (13–15) arise as the difference between the two forms above after applying thepullback by ˆ ι : N → SM and, of course, taking integrals and applying Stokes theorem. Therefore,we shall firstly study the two expressions ( P y R ξ α n ) ◦ ( B n − i ∧ i ) and P y R ξ α i and then take theirdifference in the three cases.Let us start by the remark that if T ij is skew-symmetric in i, j and P = P P k e k , then P y X ≤ i P, ν ) − (cid:18) n (cid:19) h P, ν i H (cid:17) vol N , which is the same as identity (14) before the integral over the hypersurface N with ∂N = ∅ .For i = 2, once we find n − ˆ ι ∗ (( P y R ξ α n ) ◦ ( B n − ∧ )) and ˆ ι ∗ ( P y R ξ α ) the proof will bealmost finished. Again we continue from the above formula, (17). Since ˆ ι ∗ e = 0 and e ◦ B = 0, . Albuquerque e q in position p ,ˆ ι ∗ (( P y R ξ α n ) ◦ ( B n − ∧ ))= ˆ ι ∗ n X p,j =0 n X q =1 R p jq P j ( e n ∧ · · · ∧ e p − n ∧ e q ∧ e p +1+ n ∧ · · · ∧ e n ) ◦ ( B n − ∧ )= ˆ ι ∗ n X p,j =0 n X q =1 R p jq P j X σ ∈ Sym n e n ◦ B σ ∧ · · · ∧ e q ◦ B σ p ∧ · · · ∧ e n ◦ B σ n = 2ˆ ι ∗ n X p,j =0 n X q =1 n X k =1 , k = p R p jq P j X σ ∈ Sym n : σ k = n − , σ p = n e n ◦ B ∧ · · · ∧ e n + k ∧ · · · ∧ e q ∧ · · · ∧ e n ◦ B where, applying the definition, we have assumed B = . . . = B n − = B and B n − = B n = 1,and where e n + k is in position k and e q in position p . Notice the factor 2, which accounts for thepermutations such that σ k = n, σ p = n − 1. Resuming with the computation,= 2( n − ι ∗ n X p,j =0 n X q =1 n X k =1 , k = p R p jq P j (cid:0) ( e ∧ · · · ∧ e n + k ∧ · · · ∧ e k ∧ · · · ∧ e n ) δ qk ++( e ∧ · · · ∧ e n + k ∧ · · · ∧ e p ∧ · · · ∧ e n ) δ qp (cid:1) = 2( n − ι ∗ n X p,j =0 n X k =1 , k = p (cid:0) R p jk P j e ∧ · · · ∧ e n + k ∧ · · · ∧ e k ∧ · · · ∧ e n ++ R p jp P j e ∧ · · · ∧ e n + k ∧ · · · ∧ e p ∧ · · · ∧ e n (cid:1) . Let A kb = h∇ b ν, e k i denote the matrix of the map ∇ · ν in the present frame. Then we recall ˆ ι ∗ e k = e k | N and ˆ ι ∗ e k + n = P b A kb e b | N . So the formula above yields12( n − ι ∗ (( P y R ξ α n ) ◦ ( B n − ∧ )) = n X p,j =0 n X k =1 , k = p ( − R p jk P j A kp + R p jp P j A kk )vol N = n X p,j =0 n X k =1 ( − R p jk P j A kp + R p jp P j A kk )vol N = − (cid:0) tr R ( P, ∇ · ν ) ν + nH Ric( ν, P ) (cid:1) vol N . Next we compute ˆ ι ∗ ( P y R ξ α ). First notice e jq ∧ e p + n y α == e jq ∧ e p + n y n − n X k =1 , k = p X σ : σ k = n − , σ p = n e ∧ · · · ∧ e k + n ∧ · · · ∧ e p + n ∧ · · · ∧ e n = e j ∧ n X k =1 , = p e ∧ · · · ∧ e k + n ∧ · · · ∧ e q ∧ · · · ∧ e n = (1 − δ qp ) e j ∧ e ∧ · · · ∧ e q + n ∧ · · · ∧ e q ∧ · · · ∧ e n ++ δ qp n X k =1 , = p e j ∧ e ∧ · · · ∧ e k + n ∧ · · · ∧ e p ∧ · · · ∧ e n . Albuquerque e k + n and e q in position k and p respectively. Hence for j = 0 we haveˆ ι ∗ ( P y ( e q ∧ e p + n y α )) == (1 − δ qp ) P A qp e ∧ · · · ∧ e p ∧ · · · ∧ e q ∧ · · · ∧ e n ++ δ qp P n X k =1 , = p A kk e ∧ · · · ∧ e k ∧ · · · ∧ e p ∧ · · · ∧ e n = ( − (1 − δ qp ) P A qp + δ qp P n X k =1 , = p A kk )vol N = ( − P A qp + δ qp nP H )vol N . Now we look again the previous computation and its summands when j = p and j = p . Then, for0 < j , we find that e jq ∧ e p + n y α = δ jp (1 − δ qp ) e j ∧ e ∧ · · · ∧ e q + n ∧ · · · ∧ e q ∧ · · · ∧ e n +(1 − δ jp ) δ qp e j ∧ e ∧ · · · ∧ e j + n ∧ · · · ∧ e p ∧ · · · ∧ e n = δ jp (1 − δ qp ) e q + n ∧ e ∧ · · · ∧ e q ∧ · · · ∧ e p ∧ · · · ∧ e n − (1 − δ jp ) δ qp e j + n ∧ e ∧ · · · ∧ e j ∧ · · · ∧ e p ∧ · · · ∧ e n and thus, since P is horizontal,ˆ ι ∗ ( P y ( e jq ∧ e p + n y α )) == δ jp (1 − δ qp ) (cid:0) − P A q e ··· n + P A q e ··· n − · · · − ( − q P q A qq e q ··· ( q − q +1) ··· n · · · (cid:1) − (cid:0) . . . (cid:1) = n X l =1 (cid:0) − δ jp (1 − δ qp ) P l A ql + δ qp (1 − δ jp ) P l A jl (cid:1) vol N . In sum,ˆ ι ∗ ( P y R ξ α ) = ˆ ι ∗ (cid:0) P y X ≤ j For any vector field P on M , lifted as horizontal vector field on T M , we have ( L P B ) Y = B ∇ ∗ Y P, ∀ Y ∈ T T M. (18) Proof. ∇ ∗ denotes the pullback connection either to π ∗ T M and to π ⋆ T M thus giving a metricconnection over T M . The result follows immediately from [2, Proposition 2.1], where it is foundthat for any vector field P over T M we have ( L P B ) Y = B ∇ ∗ Y P − ∇ ∗ BY P . In the present setting, P is a horizontal lift, hence a pullback to SM to the horizontal tangent subbundle. Since BY isvertical, we clearly have ∇ ∗ BY P = 0. (cid:4) The case of Theorem 2.1 with P Killing-conformal in ambient constant sectional curvature M has the first formulations in [11] and more recently in [9].We now apply the theorem to a position vector field P on a oriented Riemannian manifold M .Let us recall the definition: P is a C vector field over an open subset U ⊂ M for which there is afunction f such that ∇ X P = f X , ∀ X ∈ X U . Theorem 2.2 (Three generalized Hsiung-Minkowski identities) . Let ι : N ֒ → M be a closed orientedimmersed hypersurface of the oriented Riemannian ( n + 1) -dimensional C manifold M . Let P bea position vector field of M defined on a neighborhood of ι ( N ) . Then we have that Z N ( f − h P, ν i H ) vol = 0 (19) and Z N (cid:16) (cid:18) n (cid:19) f H − (cid:18) n (cid:19) h P, ν i H − Ric( P, ν ) + h P, ν i Ric( ν, ν ) (cid:17) vol = 0 (20) and Z N (cid:16)(cid:0) h P, ν i Ric( ν, ν ) − Ric( ν, P ) (cid:1) nH + 3 (cid:18) n (cid:19) f H − (cid:18) n (cid:19) h P, ν i H ++ h P, ν i tr( R ( ν, ∇ · ν ) ν ) + Ric( ∇ P ν, ν ) − tr( R ( P, ∇ · ν ) ν ) (cid:17) vol = 0 . (21) Proof. First, the function f on M corresponds to an obvious function on T M constant along thefibres. Then it follows that ∇ ∗ π ∗ Y π ∗ P = f π ∗ Y and, of course, ∇ ∗ π ⋆ Y π ∗ P = 0, for all Y ∈ T M .Applying the above proposition L P B = f B . And we further conclude by a proved Leibniz rule1 i !( n − i )! α n ◦ ( L P B n − i ∧ i ) = ( n − i ) f α i . Applying ˆ ι ∗ we find ( n − i ) f (cid:0) ni (cid:1) H i vol N , which is the same as ( i + 1) (cid:0) ni +1 (cid:1) f H i vol N , and the resultfollows for i = 0 , , (cid:4) . Albuquerque i = 0, may be attributed to Hsiung inthe general Riemannian setting; however, the meaning of a position vector field in [8] is difficult tograsp, being given with respect to an assumed orthogonal frame. According to [4, 11], the positionvector is parallel to the mean curvature vector field. Then implying f is always 1, which is not eventhe case for space forms. We further remark that, if the hypersurface has boundary, the right handsides of the identities in Theorems 2.1 and 2.2, in respective order of i = 0 , , 2, are R ∂N ˆ ι ∗ ( P y α i ).The following result gives some useful identities; the proof is trivial. Proposition 2.2. Let P be a position vector field and let ν denote a unit vector field on M . Thenwe have the following identities: d P ♭ = 0 L P h , i = 2 f h , i R ( , ) P = d f ∧ P, ν ) = − n d f ( ν ) tr( R ( P, ∇ · ν ) ν ) = nH d f ( ν ) . (23) In particular, if f is a constant, then R ( , ) P = 0 . Corollary 2.1. Under the hypothesis of Theorem 2.2, suppose moreover that M is an Einsteinmanifold. Then we have that Z N (cid:0) f H − h P, ν i H (cid:1) vol = 0 (24) and Z N (cid:16) (cid:18) n (cid:19) f H − (cid:18) n (cid:19) h P, ν i H + h P, ν i tr( R ( ν, ∇ · ν ) ν ) − tr( R ( P, ∇ · ν ) ν ) (cid:17) vol = 0 . (25) Proof. Suppose Ric is a constant multiple of the metric. Then clearly Ric( ∇ ν, ν ) = 0. The resultfollows by simplification of (20) and (21). (cid:4) Thus we may say the second Hsiung-Minkowski identity holds true for any Einstein metric.The third formula of Hsiung-Minkowski follows immediately for constant sectional curvatureambient manifold. Indeed, if we have R ( X, Y ) Z = c ( h Y, Z i X − h X, Z i Y ), for all X, Y, Z ∈ T M ,for some constant c , then tr R ( P, ∇ · ν ) ν = − c h P, ν i nH = h P, ν i tr( R ( ν, ∇ · ν ) ν ). And therefore, forevery hypersurface, Z N (cid:0) f H − h P, ν i H (cid:1) vol = 0 . (26)Every warped product metric admits a position vector field, cf. [2, Corollary 4.1] or [5]. Hencethe new results in this section have a large domain of application. 3. Constant mean curvatures H and H on Einstein M The applications of Hsiung-Minkowski formulas in space forms are abundant. We refer thereader to [10] for an interesting short survey on the subject. Now the use of the identities (19,24,25)within Einstein manifolds has good expectations.A particular case where the following result applies is that of isoparametric hypersurfaces, i.e.all principal curvatures constant. Theorem 3.3. Let M be an Einstein manifold admiting a position vector field P . Let N be a closedhypersurface of M and suppose H , H are constant. Then: . Albuquerque (i) If R N f vol = 0 and R N h P, ν i vol = 0 , then H = H = 0 and N is flat;(ii) If R N f vol = 0 , then none of the previous vanish and N is totally umbilic and non-flat.Proof. We start by the second case. Let a , . . . , a n denote the principal curvatures of N . 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Albuquerque Some integral formulas for closed hypersurfaces , Math. Scand. 2 (1954), 286–294.[8] C. Hsiung, Some integral formulas for closed hypersurfaces in Riemannian space , Pacific J.Math. 6 (1956), 291–299.[9] K. Kwong, An Extension of Hsiung–Minkowski Formulas and Some Applications , J. Geom.Anal. 26 (2016), 1–23.[10] K. Kwong, H. Lee and J. Pyo, Weighted Hsiung-Minkowski formulas and rigidity of umbilicalhypersurfaces , Math. Res. Lett., Volume 25 (2) (2018), 597–616.[11] Y. Katsurada, Generalized Minkowski formulas for closed hypersurfaces in Riemann space ,Ann. Mat. Pura Appl. (4), 57 (1962), 283–293. R. Albuquerque | [email protected]@uevora.pt