Optimal inequalities, contact delta-invariants and their applications

OPTIMAL INEQUALITIES, CONTACT δ-INVARIANTS AND THEIR APPLICATIONS BANG-YEN CHEN AND VER´ ONICA MART´ IN-MOLINA1 Abstract. Associated with a k-tuple (n1,...,nk)∈S(2n+ 1) with n≥1, we define a contact δ-invariant, δc(n1,...,nk), on an almost contact metric (2n+ 1)-manifold M. For an arbitrary isometric immersion of Minto a Riemannian manifold, we establish an optimal inequality involving δc(n1,...,nk) and the squared mean curvature of the immersion. Furthermore, we investigate isometric immersions of contact metric and K-contact manifolds into Riemannian space forms which verify the equality case of the inequality for some k-tuple. 1. Introduction. According to the celebrated embedding theorem of J. F. Nash [21], every Riemannian manifold can be isometrically embedded in Euclidean spaces with sufficiently high codimension. The Nash theorem was established in the hope that if Riemannian manifolds could always be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. As observed by M. Gromov [18] in 1985, this hope had not been materialized however. The main reason for this is lack of controls of the extrinsic properties of the submanifold by the known intrinsic invariants. In order to overcome such difficulties as well as to provide answers to an open question on minimal immersions, the first author introduced in the early 1990’s new types of Riemannian invariants δ(n1, . . . , nk), known as the δ-invariants (also known as Chen invariants in the literature). At the same time he was able to establish general optimal inequalities involving the new intrinsic invariants and the main extrinsic invariants, the squared mean curvature, for arbitrary Riemannian submanifolds. The δ-invariants are very different in nature from the “classical” Ricci and scalar curvatures (see [6, 7, 8]). After δ-invariants were invented and the corresponding inequalities were established in [6, 7], δ-invariants were investigated by many geometers in the last two 2000 Mathematics Subject Classification. Primary 51M16; Secondary 53C40, 53C25, 53D15. Key words and phrases. Optimal inequalities, contact δ-invariants, almost contact metric manifolds, K-contact manifolds, squared mean curvature. 1The second author is supported by the FPU scholarship program of the Ministerio de Educaci´on (Spain) and the PAI group FQM-327 (Junta de Andaluc´ıa, Spain). 1 2 B. Y. CHEN AND V. MART´ IN-MOLINA decades. Such invariants have been applied to several areas, including spectral geometry, affine geometry and general relativity (see for instance [2, 10, 11, 17, 19, 23]; in particular, see [8, 10] for a recent survey on δ-invariants and their applications). For an almost contact metric manifold Mand an integer q≥2, the first author and Mihai defined in [12] a contact invariant δc(q). They also established an optimal inequality for isometric immersions of Minto real space forms involving δ2(q). Moreover, they investigated K-contact submanifolds in Riemannian space forms which satisfy the equality case of the inequality. In this paper, we extend δc(q) to contact invariants δc(n1, . . . , nk) on almost contract metric manifolds. For an isometric immersion of an almost contract metric (2n+ 1)-manifold Minto any Riemannian manifold, we establish in section 4 an optimal inequality relating δc(n1, . . . , nk) to the squared mean curvature. In section 5, we prove a minimality result for contact metric manifolds in Riemannian space forms. In section 6, we show that K-contact manifolds in Riemannian space forms satisfying the equality case of the inequality are Sasakian. In the last section, we prove that every K-contact hypersurface of a Riemannian space form satisfying the equality case of the inequality for a k-tuple (n1, . . . , nk) with Pk j=1 nj≤2nis totally geodesic. 2. Preliminaries. We recall some general definitions and basic formulas which will be used later. For general background on almost contact metric manifolds and submanifolds, we recommend references [1] and [4] respectively. 2.1. Almost contact metric manifolds. An odd-dimensional Riemannian manifold (M, g) is called an almost contact metric manifold if there exist on Ma (1,1)- tensor field φ, a vector field ξand a 1-form ηsuch that φ2X=−X+η(X)ξ, η(ξ) = 1,(2.1) g(φX, φY ) = g(X, Y )−η(X)η(Y) (2.2) for vector fields X, Y on M. On an almost contact metric manifold, we also have φξ = 0 and η◦φ= 0. The vector field ξis called the structure vector field. By a contact manifold we mean a (2n+ 1)-manifold Mtogether with a global 1-form ηsatisfying η∧(dη)n6= 0 on M. If ηof an almost contact metric manifold (M, φ, ξ, η, g) is a contact form and if ηsatisfies dη(X, Y ) = g(X, φY ) for all vectors X, Y tangent to M, then Mis called a contact metric manifold. A contact metric manifold is called K-contact if its characteristic vector field ξis a Killing vector field. It is well-known that a contact metric manifold is a K-contact manifold if and only if ∇Xξ=−φX (2.3) OPTIMAL INEQUALITIES 3 holds for all vector fields Xon M. In fact, an almost contact metric manifold satisfying condition (2.3) is also a K-contact manifold. Condition (2.3) is equivalent to K(X, ξ) = 1 (2.4) for every unit tangent vector Xorthogonal to ξ. An almost contact metric structure of Mis called normal if the Nijenhuis torsion [φ, φ] of φequals to −2dη⊗ξ. A normal contact metric manifold is called a Sasakian manifold. It can be proved that an almost contact metric manifold is Sasakian if and only if the Riemann curvature tensor Rsatisfies R(X, Y )ξ=η(Y)X−η(X)Y(2.5) for any vector fields X, Y on M. A Sasakian manifold is also K-contact but the converse is not true in general if dim M≥5. On a contact metric (2n+1)-manifold M,η= 0 defines a 2n-dimensional distribution in TM, which is called the contact distribution. A submanifold Nof M is called an integral submanifold if η(X) = 0 for every tangent vector X∈TN. On a contact manifold of dimension 2n+ 1, there exist integral submanifolds of the contact distribution of dimension less than or equal to n, but of no higher dimension. 2.2. Basic formulas, equations and definitions. Let (M, φ, ξ, η, g) be a (2n+ 1)-dimensional almost contact metric manifold with n≥1 isometrically immersed in a Riemannian m-manifold ( ˜ Mm,˜g). Let h,idenote the inner product of ˜ Mm as well as of M. Denote by ∇and ˜ ∇the Levi-Civita connections on Mand ˜ Mmrespectively. Let h, D and Abe the second fundamental form, the normal connection, and shape operator of M, respectively. The Gauss and Weingarten formulas are given by ˜ ∇XY=∇XY+h(X, Y ),(2.6) ˜ ∇XV=−AVX+DXV(2.7) for vector fields X, Y tangent to Mand Vnormal to M. Denote by Rand ˜ Rthe Riemann curvature tensors of Mand ˜ Mm, respectively. Then the Gauss and Codazzi equations are given by hR(X, Y )Z, Wi=h˜ R(X, Y )Z, W i+hh(X, W ), h(Y, Z)i(2.8) − hh(X, Z), h(Y, W)i, (˜ R(X, Y )Z)⊥= ( ¯ ∇Xh)(Y, Z)−(¯ ∇Yh)(X, Z),(2.9) where X, Y, Z, W are tangent vectors of M, ( ˜ R(X, Y )Z)⊥is the normal component of ˜ R(X, Y )Z, and ¯ ∇his defined by (¯ ∇Xh)(Y, Z) = DXh(Y, Z)−h(∇XY, Z)−h(Y, ∇XZ).(2.10) 4 B. Y. CHEN AND V. MART´ IN-MOLINA When the ambient space ˜ Mmis a Riemannian space form of constant curvature c, equations (2.8) and (2.9) of Gauss and Codazzi reduce to hR(X, Y )Z, Wi=c{hX, W i hY, Zi−hX, Zi hY, W i} (2.11) +hh(X, W), h(Y, Z)i−hh(X, Z), h(Y, W)i, (¯ ∇Xh)(Y, Z)=(¯ ∇Yh)(X, Z),(2.12) The mean curvature vector of Min ˜ Mmis defined by H= (trace h)/dim M. The squared mean curvature H2is defined as H2=hH, Hi. 2.3. Twisted products. The notion of twisted products was introduced in [5, page 66] as follows: Let Band Fbe Riemannian manifolds with Riemannian metrics gBand gF, respectively, and fa positive differentiable function on B×F. Consider the product manifold B×Fwith its projection πB:B×F→Band πF:B×F→F. The twisted product B×fFis the manifold B×Fequipped with the Riemannian structure such that ||X||2=||πB∗(X)||2+f2||πF∗(X)||2 for any vector Xtangent to B×fF. Thus, we have g=gB+f2gF. The function fabove is called the twisting function of the twisted product. When fdepends only on B, the twisted product is a warped product and fis called the warping function. For vector fields V, X tangent to Band Frespectively, we have ∇XV=∇VX= (Xf)V, (2.13) where ∇is the Levi-Civita connection of the twisted product B×fF. 3. Contact δ-invariants. Suppose that (M, φ, ξ, η, g) is an almost contact metric (2n+ 1)-manifold with n≥1. If {e1, . . . , er}is an orthonormal basis of a linear r-space L⊆TpMat p∈M, we define the scalar curvature of Lby τ(L) = X 1≤α<β≤r K(eα, eβ). The scalar curvature τ(p) at a point p∈Mis τ(p) = τ(TpM). For an integer k≥1, let S(2n+1, k) be the set of k-tuples (n1, . . . , nk) of integers satisfying 2≤n1,· · · , nk≤2nand 2 ≤n1+· · · +nk≤2n+ 1. We denote by S(2n+ 1) the union: ∪k≥1S(2n+ 1, k). For each k-tuple (n1, . . . , nk)∈ S(2n+ 1), the δ-invariant δ(n1, . . . , nk) was introduced in [7]. Now, we define the contact version of the δ-invariant in the same spirit as in [7, 12]. OPTIMAL INEQUALITIES 5 Definition 3.1. Let Mbe an almost contact metric (2n+1)-manifold and (n1, . . . , nk)∈ S(2n+ 1). The contact δ-invariant δc(n1, . . . , nk) is defined by δc(n1, . . . , nk)(p) := τ(p)−inf ©τ(L1) + ···+τ(Lk)ª, where L1, L2, . . . , Lkrun over all mutually orthogonal subspaces of TpMso that L1 contains the characteristic vector ξand dim Lj=nj,j= 1, . . . , k. Definition 3.2. Let Mbe an almost contact metric manifold and let L1, . . . , Lk be mutually orthogonal subspaces of TpMwith dim Lj≥2, j= 1, . . . , k. A plane section π⊂TpMis said to be orthogonal to L1, . . . , Lkif there exists an orthonormal basis {¯e1,¯e2}such that π= Span{¯e1,¯e2}and one of the following three cases occurs: (1) ¯e1∈Liand ¯e2∈Ljwith 1 ≤i6=j≤k; (2) ¯e1∈Lifor some i∈ {1, . . . , k}and ¯e2⊥L1, . . . , Lk; (3) ¯e1,¯e2⊥L1, . . . , Lk. We call an orthonormal frame {e1, . . . , e2n+1}an orthonormal ξ-frame if e1is parallel to ξ. We need the following algebraic lemma from [6] for later use. Lemma 3.1. Let a1, . . . , ap, ζ be p+ 1 real numbers such that Ãp X i=1 ai!2 = (p−1)Ãζ+ p X i=1 a2 i!. Then 2a1a2≥ζ, with equality holding if and only if a1+a2=a3=. . . =ap. 4. An optimal inequality. For each k-tuple (n1, . . . , nk)∈ S(2n+ 1), let c(n1, . . . , nk) and b(n1, . . . , nk) be the positive numbers given by c(n1, . . . , nk) = (2n+ 1)2(2n+k−Pk j=1 nj) 2(2n+k+ 1 −Pk j=1 nj), b(n1, . . . , nk) = n(2n+ 1) −1 2 k X j=1 nj(nj−1). Put ∆1={1, . . . , n1}, . . . , ∆k={n1+· · · +nk−1+ 1, . . . , n1+· · · +nk}, ∆ = ∆1∪ · · · ∪ ∆k,∆2= (∆1×∆1)∪ · · · ∪ (∆k×∆k). Now, we modify the proof of [9, Theorem 3.1] to obtain the following. Theorem 4.1. Let Mbe a (2n+ 1)-dimensional almost contact metric manifold isometrically immersed in a Riemannian m-manifold ˜ Mm. Then, for each point 6 B. Y. CHEN AND V. MART´ IN-MOLINA p∈Mand each k-tuple (n1, . . . , nk)∈ S(2n+ 1), we have: δc(n1, . . . , nk)(p)≤c(n1, . . . , nk)H2(p) + b(n1, . . . , nk) max ˜ K(p),(4.1) where max ˜ K(p)is the maximum of the sectional curvature function of ˜ Mmrestricted to 2-plane sections of the tangent space TpM. Moreover, the equality case of inequality (4.1) holds at pif and only the following two conditions hold: (a) There exists an orthonormal ξ-basis {e1, . . . , e2n+1}of TpMand an orthonormal basis {e2n+2, . . . , em}of the normal space T⊥ pMsuch that the shape operator with respect to {e1, . . . , em}satisfies Aer=      Ar 1. . . 0 . . ..... . .0 0. . . Ar k 0µrI       , r = 2n+ 2, . . . , m, (4.2) where Iis an identity matrix and each Ar jis a symmetric nj×njsubmatrix satisfying trace (Ar 1) = · · · = trace (Ar k) = µr.(4.3) (b) There exist mutually orthogonal subspaces L1, . . . , Lkof TpMwith ξ∈L1 and δc(n1, . . . , nk)(p) = τ(p)−Pk j=1 τ(Lj)such that any plane section π⊂TpM orthogonal to L1, . . . , Lksatisfies ˜ K(π) = max ˜ K(p). Proof. Let Mbe an almost contact metric (2n+1)-manifold isometrically immersed in ˜ Mm. Then, at a point p∈M, the equation of Gauss gives 2τ(p) = (2n+ 1)2H2(p)− ||h||2(p) + 2˜τ(TpM),(4.4) where ||h||2is the squared norm of hand ˜τ(TpM) is the scalar curvature of the ambient space ˜ Mmcorresponding to TpM⊂Tp˜ Mm. Let us put η= 2τ(p)−(2n+ 1)2(2n+k−Pk j=1 nj) 2n+k+ 1 −Pk j=1 nj H2(p)−2˜τ(TpM).(4.5) Then (4.4) and (4.5) give (2n+ 1)2H2(p) = γ¡η+||h||2(p)¢, γ = 2n+k+ 1 −n1− · · · − nk.(4.6) Let us choose an orthonormal ξ-basis {e1, . . . , e2n+1}of TpMin such way that eαi∈Lifor each αi∈∆i. We choose an orthonormal basis {e2n+2, . . . , em}of T⊥ pMso that e2n+2 in the direction of Hat p. Then (4.6) yields ³X2n+1 A=1 aA ´2=γhη+X A a2 A+X A6=B (h2n+2 AB )2+ m X r=2n+3 2n+1 X A,B=1 (hr AB)2i(4.7) OPTIMAL INEQUALITIES 7 with aA=h2n+2 AA for 1 ≤A, B ≤2n+ 1. Equation (4.7) is equivalent to ³Xγ+1 i=1 ¯ai´2=γhη+ γ+1 X i=1 (¯ai)2+X A6=B (h2n+2 AB )2+ m X r=2n+3 2n+1 X A,B=1 (hr AB)2 −X 2≤α1<β1≤n1 2aα1aβ1−X α2<β2 2aα2aβ2− · · · − X αk<βk 2aαkaβki, (4.8) where α2, β2∈∆2, . . . , αk, βk∈∆kand ¯a1=a1,¯a2=a2+· · · +an1, ¯a3=an1+1 +· · · +an1+n2, . . . ¯ak+1 =an1+···+nk−1+1 +· · · +an1···+nk, ¯ak+2 =an1···+nk+1, . . . , ¯aγ+1 =a2n+1. Thus, by applying Lemma 3.1 to (4.8) we obtain X 1≤α1<β1≤n1 aα1aβ1+X α2<β2 aα2aβ2+· · · +X αk<βk aαkaβk ≥η 2+X A<B (h2n+2 AB )2+ m X r=2n+3 2n+1 X A,B=1 1 2(hr AB)2, (4.9) where αi, βi∈∆i, i = 1, . . . , k. It also follows from Lemma 3.1 that the equality sign of (4.9) holds if and only if a1+ ¯a2= ¯a3=· · · = ¯aγ+1. On the other hand, equation of Gauss implies that, for each j∈ {1, . . . , k}, τ(Lj) = m X r=2n+2 X αj<βj³hr αjαjhr βjβj−(hr αjβj)2´+ ˜τ(Lj)(4.10) for αj, βj∈∆j, where ˜τ(Lj) is the scalar curvature of Ljin ˜ Mm. Then, by combining (4.5), (4.9) and (4.10) we obtain τ(L1) + · · · +τ(Lk)≥η 2+1 2 m X r=2n+2 X (α,β)/∈∆2 (hr αβ)2 +1 2 m X r=2n+3 k X j=1 ³X αj∈∆j hr αjαj´2+ k X j=1 ˜τ(Lj)≥η 2+ k X j=1 ˜τ(Lj) =τ−(2n+ 1)2(2n+k−Pnj) 2(2n+k+ 1 −Pnj)H2−˜τ(TpM) + k X j=1 ˜τ(Lj). (4.11) From (4.11) we find τ− k X j=1 τ(Lj)≤(2n+ 1)2(2n+k−Pnj) 2(2n+k+ 1 −Pnj)H2+ ˜τ(TpM)− k X j=1 ˜τ(Lj),(4.12) 8 B. Y. CHEN AND V. MART´ IN-MOLINA which implies that δc(n1, . . . , nk)≤(2n+ 1)2(2n+k−Pnj) 2(2n+k+ 1 −Pnj)H2+e δc(n1, . . . , nk),(4.13) where e δc(n1, . . . , nk) := ˜τ(TpM)−inf{˜τ(˜ L1) + · · · + ˜τ(˜ Lk)}(4.14) with ˜ L1, . . . , ˜ Lkrun over all kmutually orthogonal subspaces of TpMsuch that ξ∈˜ L1and dim ˜ Lj=nj;j= 1, . . . , k. From (4.13) we obtain (4.1). If the equality case of (4.1) holds at p, then all of the inequalities in (4.9) and (4.11) become equalities. Hence, we obtain condition (a). Moreover, it follows from (4.13) and (4.14) that condition (b) holds too. The converse can be easily verified. ¤ As an immediate consequence of Theorem 4.1, we have the following. Theorem 4.2. Let Mbe an almost contact metric (2n+ 1)-manifold isometrically immersed in a Riemannian space form Rm(c)of constant curvature c. Then, for any k-tuple (n1, . . . , nk)∈ S(2n+ 1), we have: δc(n1, . . . , nk)≤c(n1, . . . , nk)H2+b(n1, . . . , nk)c. (4.15) The equality case of inequality (4.15) holds at a point p∈Mif and only if there exists an orthonormal ξ-basis {e1, . . . , e2n+1}of TpMand an orthonormal basis {e2n+2, . . . , em}of T⊥ pMsuch that the shape operator with respect to {e1, . . . , em} satisfies Aer=      Ar 1. . . 0 . . ..... . .0 0. . . Ar k 0µrI       , r = 2n+ 2, . . . , m, (4.16) where each Ar jis a symmetric nj×njsubmatrix satisfying trace (Ar 1) = · · · = trace (Ar k) = µr.(4.17) Definition 4.1. Let Mbe an almost contact metric (2n+ 1)-manifold immersed in ˜ Mm. If Msatisfies the equality case of (4.1) for a k-tuple (n1, . . . , nk), then an orthonormal ξ-frame {e1, . . . , e2n+1}satisfying (4.2) and (4.3) is called an adapted ξ-frame. 5. Minimality. Let Mbe an almost contact metric (2n+ 1)-manifold isometrically immersed in a Riemannian m-manifold ˜ Mm. If Msatisfies the equality case of (4.1) for some OPTIMAL INEQUALITIES 9 (n1, . . . , nk)∈ S(2n+1). Then, with respect to an adapted ξ-frame {e1, . . . , e2n+1}, we have h(eα, Y ) = (2n+ 1) heα, Y i 2n+k+1−Pk j=1nj H, ∀Y∈TM, (5.1) for α∈ {1 + Pk j=1 nj, . . . , 2n+ 1}. Next, we prove the following minimality result. Theorem 5.1. Let Mbe a contact metric (2n+1)-manifold isometrically immersed in a Riemannian space form Rm(c). If Msatisfies the equality case of (4.15) for a k-tuple (n1, . . . , nk)∈ S(2n+ 1) with Pk j=1 nj≤n, then Mis a minimal submanifold of Rm(c) Proof. Let Mbe a contact metric (2n+ 1)-manifold isometrically immersed in Rm(c) such that the equality case of (4.15) is satisfied for some k-tuple (n1, . . . , nk) with Pk j=1 nj≤n. Then Theorem 4.2 implies that there exists an adapted ξ-basis {e1, . . . , e2n+1}of TpMand an orthonormal basis {e2n+2, . . . , em}of T⊥ pMsuch that (4.16) and (4.17) hold. Now, assume that Mis non-minimal in Rm(c), i.e. H6= 0. In order to derive a contradiction, let us put D(p)=(X∈TpM:h(X, Y ) = (2n+ 1) hX, Y i 2n+k+1−Pk j=1nj H, ∀Y∈TpM). It follows from (5.1) that dim D(p)≥2n+1−Pk j=1 nj. Clearly, dim D(p) is constant on some nonempty open submanifold, say U, of M. Since H6= 0, it follows from the definition of adapted ξ-frame, (4.16) and (4.17) that, for any X∈ D, we have η(X) = 0. Hence, Dis a contact distribution on U⊂M. Let D⊥be the orthogonal complementary distribution of D. Then we have h(D,D⊥) = {0}. Thus, for vector fields X, Y ∈ D and Z∈ D⊥, we have (¯ ∇Xh)(Y, Z) = −h(∇XY, Z)−h(Y, ∇XZ). Therefore, after applying the equation of Codazzi, we obtain h([X, Y ], Z) = h(∇XY, Z)−h(∇YX, Z) =−(¯ ∇Xh)(Y, Z)−h(∇YX, Z)−h(Y, ∇XZ) =−(¯ ∇Yh)(X, Z)−h(∇YX, Z)−h(Y, ∇XZ) =h(X, ∇YZ)−h(Y, ∇XZ) =2n+ 1 2n+k+1−Pk j=1nj (hX, ∇YZi−hY, ∇XZi)H =2n+ 1 2n+k+1−Pk j=1nj h[X, Y ], ZiH, which implies [X, Y ]∈ D. Hence, Dis an involutive distribution on Uwhose leaves are integral submanifolds of M. 16 B. Y. CHEN AND V. MART´ IN-MOLINA [10] B. Y. Chen, δ-invariants, inequalities of submanifolds and their applications, in: Topics in Differential Geometry, Editura Academiei Romˆane, Bucharest, (eds. A. Mihai, I. Mihai and R. Miron) (2008). [11] B. Y. Chen, F. Dillen and L. Verstraelen, δ-invariants and their applications to centroaffine geometry, Differential Geom. Appl. 22 (2005), 341–354. [12] B. Y. Chen and I. Mihai. 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Department of Mathematics, Michigan State University, East Lansing, Michigan, 48824–1027, U.S.A. E-mail address:[email protected] Departamento de Geometr´ ıa y Topolog´ ıa, Facultad de Matem´ aticas, Universidad de Sevilla, Apartado de Correos 1160, 41080 Sevilla, Spain. E-mail address:[email protected]