Y. J. Suh and H. Lee (Bull. Korean. Math. Soc. 47, 551-561 (2010)) characterized real hypersurfaces $M$ of type $B$ by the invariance of vector bundle $JTM^\perp$ under the shape operator and the orthogonality of $JTM^\perp$ and $\mathcal {J}TM^\perp$, where $TM^\perp$, $J$ and $\mathcal J$ are the normal bundle of $M$, Kähler structure and Quaternionic Kähler structure of $G_2({\mathbb{C}}^{m+2})$ respectively. In this paper, we characterize real hypersurfaces $M$ of type A by the invariance of the vector bundle $JTM^\perp$ under the shape operator with the Reeb vector field in $\mathcal {J}TM^\perp$.
In this paper we give a characterization of real hypersurfaces in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, $m \geq 2$ with Reeb vector field $ξ$ belonging to the maximal quaternionic subbundle $\mathcal Q$. Then it becomes a tube over a totally real totally geodesic ${\mathbb H}H^n$, $m=2n$, in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, a horosphere whose center at the infinity is singular or another exceptional case.
There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of generalized Tanaka-Webster Reeb parallel Ricci tensor for $M$ in $G_2({\mathbb C}^{m+2})$. By using such parallel conditions, we give complete classifications of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$.
Using generalized Tanaka-Webster connection, we considered a real hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ when the GTW Reeb Lie derivative of the structure Jacobi operator coincides with the Reeb Lie derivative. Next using the method of simultaneous diagonalization, we prove a complete classification for a real hypersurface in $G_2({\mathbb C}^{m+2})$ satisfying such a condition. In this case, we have proved that $M$ is an open part of a tube around a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.
We classify all of real hypersurfaces $M$ with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m \geq 2$. Then it becomes a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a horosphere whose center at infinity is singular and of type $JN \in {\mathfrak J}N$ for a unit normal vector field $N$ of $M$.
We investigate the structure of real hypersurfaces with isometric Reeb flow in Kaehler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type.
We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.
We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Q^m for m > 2. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space CP^k which is embedded canonically in Q^{2k} as a totally geodesic complex submanifold. As a consequence we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics.
In this paper we study real hypersurfaces in the complex quadric space $Q^m$ whose structure Jacobi operator commutes with their structure tensor field. We show that the Reeb curvature $α$ of such hypersurfaces is constant and if $α$ is non-zero then the hypersurface is a tube around a totally geodesic submanifold $\mathbb{C} P^k \subset Q^m$, where $m=2k$. We also consider Reeb flat hypersurfaces, namely, when the Reeb curvature is zero. We show that the tube around $\mathbb{C} P^k \subset Q^m$ ($m=2k$), with radius $\fracπ{4}$ is the only Reeb flat Hopf hypersurface with commuting Ricci tens…
In this paper, we first introduce the full express of the Riemannian curvature tensor of a real hypersurface $M$ in complex quadric $Q^{m}$ from the equation of Gauss. Next we derive a formula for the structure Jacobi operator $R_ξ$ and its derivative under the Levi-Civita connection of $M$. We give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, $\nabla_ξR_ξ =0$, in the complex quadric $Q^{m}$, $m \geq 3$.
It is proved the non-existence of Hopf hypersurfaces in $G_{2}({\Bbb C}^{m+2})$, $m \geq 3$, whose normal Jacobi operator is semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing and the component of the Reeb vector field in the maximal quaternionic subbundle ${\frak D}$ or its orthogonal complement ${\frak D}^{\bot}$ is invariant by the shape operator.
We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics ${Q^*}^{m} = SO^{o}_{2,m}/SO_mSO_2$, $m \geq 3$. We show that $m$ is even, say $m = 2k$, and any such hypersurface becomes an open part of a tube around a $k$-dimensional complex hyperbolic space ${\mathbb C}H^k$ which is embedded canonically in ${Q^*}^{2k}$ as a totally geodesic complex submanifold or a horosphere whose center at infinity is $\frak A$-isotropic singular. As a consequence of the result, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex qua…
We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.
We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the nonexistence of real hypersurface in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.
In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $δ$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hold.
In this paper, we introduce a notion of quadratic Killing structure Jacobi operator (simply, Killing structure Jacobi operator) and its geometric meaning for real hypersurfaces in the complex two-plane Grassmannians. In addition, we give a classification theorem for Hopf real hypersurfaces with quadratic Killing structure Jacobi operator in complex two-plane Grassmannians.
The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassmannians. This answers a question considered in [Monatsh Math, 172 (2013), 167-178] in negative.
In this paper, we have introduced a new notion of generalized Tanaka-Webster Reeb recurrent Ricci tensor in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Next, we give a non-existence property for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with such a condition.
In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_ξϕT=TR_ξϕ$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, we give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition respectively.
In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$.
In this paper, we have considered a new commuting condition, that is, $(R_ξϕ) S = S (R_ξϕ)$ \big(resp. $(\Bar{R}_Nϕ) S = S (\Bar{R}_Nϕ$)\big) between the restricted Jacobi operator~$R_ξϕ$ (resp. $\Bar{R}_Nϕ$), and the Ricci tensor $S$ for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$. In terms of this condition we give a complete classification for Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$.
In \cite{S 2017}, Suh gave a non-existence theorem for Hopf real hypersurfaces in the complex quadric with parallel normal Jacobi operator. Motivated by this result, in this paper, we introduce some generalized conditions named $\mathcal C$-parallel or Reeb parallel normal Jacobi operators. By using such weaker parallelisms of normal Jacobi operator, first we can assert a non-existence theorem of Hopf real hypersurfaces with $\mathcal C$-parallel normal Jacobi operator in the complex quadric $Q^{m}$, $m \geq 3$. Next, we prove that a Hopf real hypersurface has Reeb parallel normal Jacobi opera…
In this paper we first introduce the full expression of the curvature tensor of a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m{\ge}2$ from the equation of Gauss. Next we derive a new formula for the Ricci tensor of $M$ in $SU_{2,m}/S(U_2{\cdot}U_m)$. Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$ with commuting Ricci tensor. Each can be described as a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a h…
We investigate integrable second order equations of the form F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0. Familiar examples include the Boyer-Finley equation, the potential form of the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota equation, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian.…
A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex hypersurfaces satisfying suitable pinching conditions and for antipodal invariant convex hypersurfaces respectively. In this work we present a generalization of these results using contact homology and a notion of dynamical convexity first introduced by Hofer-Wysocki-Zehnder for tight contact forms…
We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Ricci tensor is parallel with respect to the generalized Tanaka-Webster connection.
It was proved in Chen's paper \cite{chen} that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$ δ(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and $δ(2)$ is a $δ$-invariant introduced by him. In this paper, we study non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field. We describe how to obtain all such hypersurfaces.
We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations for the almost contact metric structure. To make the theory meaningful, we construct new families of non-degenerate Hopf real hypersurfaces whose shape operator is diagonalisable, and one Hopf example with degenerate metric and non-diagonalisable shape operator. Next, we obtain a rigidity result. We classify those real hypersurfaces which are $η$-umbilical.…
This paper gives an example of special Lagrangian manifold obtained from a hypersurface of a complex Grassmannian with vanishing first Chern class. The obtained manifold is a 1-torus bundle over the two dimensional real projective space. Such manifolds are interesting for mirror symmetry theory. Other examples of the same type are provided at the end of this article.
We study curvature-adapted submanifolds of general symmetric spaces. We generalize Cartan's theorem for isoparametric hypersurfaces of spheres and Wang's classification of isoparametric Hopf hypersurfaces in complex projective spaces to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. Various classification results in the Cayley projective and hyperbolic planes and in complex two-plane Grassmannians are obtained under some additional assumptions.
We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motivated by the work of Eliashberg and Hofer on unseen symplectic boundaries, we try to extend the above result to other classes of domains in (co)tangent bundles of surfaces. The domains we can treat consist of those bounded by deformations of negatively curved unit cotangent bundle…
Ricci soliton contact metric manifolds with certain nullity conditions have recently been studied by Ghosh and Sharma. Whereas the gradient case is well-understood, they provided a list of candidates for the nongradient case.These candidates can be realized as Lie groups, but one only knows the structures of the underlying Lie algebras, which are hard to be analyzed apart from the three-dimensional case. In this paper, we study these Lie groups with dimension greater than three, and prove that the connected, simply-connected, and complete ones can be realized as homogeneous real hypersurfaces …
We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat real hypersurface. We rely on a finiteness theorem for the space of square integrable holomorphic 2-forms on the complement of the Levi-flat real hypersurface, where the curvature plays the role of the size of the infinitesimal holonomy of its Levi foliation.
It is very well known that Hopf real hypersurfaces in the complex projective space can be locally characterized as tubes over complex submanifolds. This also holds true for some, but not all, Hopf real hypersurfaces in the complex hyperbolic space. The main goal of this paper is to show, in a unified way, how to construct Hopf real hypersurfaces in the complex hyperbolic space from a horizontal submanifold in one of the three twistor spaces of the indefinite complex $2$-plane Grassmannian with respect to the natural para-quaternionic Kähler structure. We also identify these twistor spaces with…
Let V be a real hypersurface of class C^k, k>=3, in a complex manifold M of complex dimension n+1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let θbe a contact form for (V,HT(V)), ξ_0 the Reeb vector field determined by θand assume that ξ_0 is of class C^k. In this paper we prove the following theorem (cf. Theorem 4.1): if the integral curves of ξ_0 are real analytic then there exist an open neighbourhood N\subset M of V and a solution u\in C^k(N) of the complex Monge-Ampère equation (dd^c u)^(n+1)=0 on N which is a defining equation for V. Moreover, the Mo…
For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ with $n \geq 3$, in terms of what we have called {\it generalised complex orthogonal Stiefel manifolds} of $\mathbb{C}^{n+1}$. In particular, the twistor space $\mathfrak{Z}(\mathbb{S}^{6})$ is biholomorphic…
We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest $L^2$-norm among all Sasakian metrics of fixed volume that can represent vector fields in the cone. We use links of isolated hypersurface singularities to produce examples of manifolds of Sasaki type, many of these in dimension f…
We classify 4-dimensional austere submanifolds in Euclidean space ruled by 2-planes. The algebraic possibilities for second fundamental forms of an austere 4-fold M were classified by Bryant, falling into three types which we label A, B, and C. We show that if M is 2-ruled of Type A, then the ruling map from M into the Grassmannian of 2-planes in R^n is holomorphic, and we give a construction for M starting with a holomorphic curve in an appropriate twistor space. If M is 2-ruled of Type B, then M is either a generalized helicoid in R^6 or the product of two classical helicoids in R^3. If M is…
