Hermitian–Grassmannian Submanifolds

Frontmatter

Constant Mean Curvature Spacelike Hypersurfaces in Spacetimes with Certain Causal Symmetries

Abstract

The role of some causal symmetries of spacetime which naturally arise in General Relativity is discussed. The importance of spacelike hypersurfaces of constant mean curvature (CMC) in the study of the Einstein equation is recalled. In certain spacetimes with symmetry defined by a timelike gradient conformal vector field or by a lightlike parallel vector field, uniqueness theorems of complete CMC spacelike hypersurfaces are given. In several cases, results of Calabi–Bernstein type are obtained as an application.

Alfonso Romero

Sequences of Maximal Antipodal Sets of Oriented Real Grassmann Manifolds II

Abstract

Chen–Nagano introduced the notion of antipodal sets of compact Riemannian symmetric spaces. The author showed a correspondence between maximal antipodal sets of oriented real Grassmann manifolds and certain families of subsets of finite sets and reduced the classifications of maximal antipodal sets of oriented real Grassmann manifolds to a certain combinatorial problem in a previous paper. In this paper we construct new sequences of maximal antipodal sets from those obtained in previous papers and estimate the cardinalities of antipodal sets.

Hiroyuki Tasaki

Derivatives on Real Hypersurfaces of Non-flat Complex Space Forms

Abstract

Let M be a real hypersurface of a nonflat complex space form, that is, either a complex projective space or a complex hyperbolic space. On M we have the Levi-Civita connection and for any nonnull real number k the corresponding generalized Tanaka-Webster connection. Therefore on M we consider their associated covariant derivatives, the Lie derivative and, for any nonnull k, the so called Lie derivative associated to the generalized Tanaka-Webster connection and introduce some classifications of real hypersurfaces in terms of the concidence of some pairs of such derivations when they are applied to the shape operator of the real hypersurface, the structure Jacobi operator, the Ricci operator or the Riemannian curvature tensor of the real hypersurface.

Juan de Dios Pérez

Maximal Antipodal Subgroups of the Automorphism Groups of Compact Lie Algebras

Abstract

We classify maximal antipodal subgroups of the group \(\mathrm {Aut}(\mathfrak {g})\) of automorphisms of a compact classical Lie algebra \(\mathfrak {g}\). A maximal antipodal subgroup of \(\mathrm {Aut}(\mathfrak {g})\) gives us as many mutually commutative involutions of \(\mathfrak {g}\) as possible. For the classification we use our former results of the classification of maximal antipodal subgroups of quotient groups of compact classical Lie groups. We also use canonical forms of elements in a compact Lie group which is not connected.

Makiko Sumi Tanaka, Hiroyuki Tasaki

A Nearly Kähler Submanifold with Vertically Pluri-Harmonic Lift

Abstract

We consider a certain lift from an almost Hermite submanifold to the bundle of partially complex structures of the ambient manifold. In particular, nearly Kähler submanifolds in Euclidean spaces such that the lifts are vertically pluri-harmonic are studied.

Kazuyuki Hasegawa

The Schwarz Lemma for Super-Conformal Maps

Abstract

A super-conformal map is a conformal map from a two-dimensional Riemannian manifold to the Euclidean four-space such that the ellipse of curvature is a circle. Quaternionic holomorphic geometry connects super-conformal maps with holomorphic maps. We report the Schwarz lemma for super-conformal maps and related results.

Katsuhiro Moriya

Reeb Recurrent Structure Jacobi Operator on Real Hypersurfaces in Complex Two-Plane Grassmannians

Abstract

In (Jeong et al., Acta Math Hungar 122(1–2), 173–186, 2009) [7], Jeong, Pérez, and Suh verified that there does not exist any connected Hopf hypersurface in complex two-plane Grassmannians with parallel structure Jacobi operator. In this paper, we consider more general notions as Reeb recurrent or \(\mathscr {Q}^{\bot }\)-recurrent structure Jacobi operator. By using these general notions, we give some new characterizations of Hopf hypersurfaces in complex two-plane Grassmannians.

Hyunjin Lee, Young Jin Suh

Hamiltonian Non-displaceability of the Gauss Images of Isoprametric Hypersurfaces (A Survey)

Abstract

This is a survey of the joint work [13] (Bull Lond Math Soc 48(5), 802–812, 2016) with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.). The Floer homology of Lagrangian intersections is computed in few cases. Here, we take the image \(L=\mathscr {G}(N)\) of the Gauss map of isoparametric hypersurfaces N in \(S^{n+1}\), that are minimal Lagrangian submanifolds of the complex hyperquadric \(Q^n(\mathbb {C})\). We call L Hamiltonian non-displaceable if \(L\cap \varphi (L)\ne \emptyset \) holds for any Hamiltonian deformation \(\varphi \). Hamiltonian non-displaceability is needed to define the Floer homology \({ HF}(L)\), since \({ HF}(L)\) is generated by points in \(L\cap \varphi (L)\). We prove the Hamiltonian non-displaceability of \(L=\mathscr {G}(N)\) for any isoparametric hypersurfaces N with principal curvatures having plural multiplicities. The main result is stated in Sect. 4.

Reiko Miyaoka

Counterexamples to Goldberg Conjecture with Reversed Orientation on Walker 8-Manifolds of Neutral Signature

Abstract

The famous Goldberg conjecture (Goldberg, Proc Am Math Soc 21, 96–100, 1969) [8] states that the almost complex structure of a compact almost-Kähler Einstein Riemannian manifold is Kähler. It is true if the scalar curvature of the manifold is nonnegative (Sekigawa, Math Ann 271, 333–337, 1985) [20], (Sekigawa, J Math Soc Jpn 36, 677–684, 1987) [21]. If we turn our attention to indefinite metric spaces, several counterexamples to the conjecture have been reported (cf. (Matsushita, J Geom Phys 55, 385–398, 2005) [17], (Matsushita et al., Monatsh Math 150, 41–48, 2007) [18], (Matsushita, et al., Proceedings of The 19th International Workshop on Hermitian-Grassmannian Submanifolds and Its Applications and the 10th RIRCM-OCAMI Joint Differential Geometry Workshop, Institute for Mathematical Sciences (NIMS), vol 19, pp 1–14. Daejeon, South Korea, 2015) [19]). It is important to recognize that all known counterexamples to date are constructed on Walker manifolds, equipped with an almost complex structure of normal orientation. In the present paper, we focus our attention on Walker manifolds with an opposite almost complex structure, and consider if counterexamples to the Goldberg conjecture can be constructed. We succeeded in finding such a counterexample on an 8-dimensional compact Walker manifold of neutral signature, but failed in the case of 6-dimensional compact Walker manifold of signature (4, 2) with a canonically defined opposite almost complex structure.

Yasuo Matsushita, Peter R. Law

A Construction of Weakly Reflective Submanifolds in Compact Symmetric Spaces

Abstract

In this paper, we give sufficient conditions for orbits of Hermann actions to be weakly reflective in terms of symmetric triads, that is a generalization of irreducible root systems. Using these sufficient conditions, we obtain new examples of weakly reflective submanifolds in compact symmetric spaces.

Shinji Ohno

Dual Orlicz Mixed Quermassintegral

Abstract

We study the dual Orlicz mixed Quermassintegral. For arbitrary monotone continuous function \(\phi \), the dual Orlicz radial sum and dual Orlicz mixed Quermassintegral are introduced. Then the dual Orlicz–Minkowski inequality and dual Orlicz–Brunn–Minkowski inequality for dual Orlicz mixed Quermassintegral are obtained. These inequalities are just the special cases of their \(L_p\) analogues (including cases \(-\infty<p<0\), \(p=0\), \(0<p<1\), \(p=1\), and \(1<p<+\infty \)). These inequalities for \(\phi =\log t\) are related to open problems including log-Minkowski problem and log-Brunn-Minkowski problem. Moreover, the equivalence of the dual Orlicz–Minkowski inequality for dual Orlicz mixed Quermassintegral and dual Orlicz–Brunn–Minkowski inequality for dual Orlicz mixed Quermassintegral is shown.

Jia He, Denghui Wu, Jiazu Zhou

Characterizations of a Clifford Hypersurface in a Unit Sphere

Abstract

The Clifford hypersurface is one of the simplest compact hypersurfaces in a unit sphere. We give two different characterizations of Clifford hypersurfaces among constant m-th order mean curvature hypersurfaces with two distinct principal curvatures. One is obtained by assuming embeddedness and by comparing two distinct principal curvatures. The proof uses the maximum principle to the two-point function, which was used in the proof of Lawson conjecture by Brendle (Acta Math. 211(2):177–190, 2013, [6]). The other is given by obtaining a sharp curvature integral inequality for hypersurfaces in a unit sphere with constant m-th order mean curvature and with two distinct principal curvatures, which generalizes Simons integral inequality (Simons, Ann. Math. (2) 88:62–105, 1968, [30]). This article is based on joint works (Min and Seo, Math. Res. Lett. 24(2):503–534, 2017, [18], Min and Seo, Monatsh. Math. 181(2):437–450, 2016, [19]) with Sung-Hong Min.

Keomkyo Seo

3-Dimensional Real Hypersurfaces with -Harmonic Curvature

Abstract

We classify real hypersurfaces with \(\eta \)-harmonic curvature of a non-flat complex space form of complex dimension 2 under the condition that the Ricci tensor S satisfies \(S\xi =\beta \xi \) where \(\beta \) is a function and \(\xi \) is the structure vector field.

Mayuko Kon

Gromov–Witten Invariants on the Products of Almost Contact Metric Manifolds

Abstract

We investigate Gromov–Witten invariants and quantum cohomologies on the products of almost contact metric manifolds. The product of two cosymplectic manifolds has a Kähler structure. We compute some cohomology classes of compact cosymplectic manifolds and show that any compact simply connected Kähler manifold cannot be a product of two cosymplectic manifolds. On the products we get some geometric properties, Gromov–Witten invariants and quantum cohomologies. We have some relations between Gromov–Witten invariants of the products and the ones of two cosymplectic manifolds.

Yong Seung Cho

On LVMB, but Not LVM, Manifolds

Abstract

The aim of this paper is to survey the constructions of the so-called LVM or LVMB manifolds after López de Medrano, Verjovsky, Meersseman, and Bosio, and to discuss some recent results as well as interesting related open questions.

Jin Hong Kim

Inequalities for Algebraic Casorati Curvatures and Their Applications II

Abstract

Different kind of algebraic Casorati curvatures are introduced. A result expressing basic Casorati inequalities for algebraic Casorati curvatures is presented and equality cases are discussed. As their applications, basic Casorati inequalities for different \(\delta \)-Casorati curvatures for different kind of submanifolds of quaternionic space forms are presented.

Young Jin Suh, Mukut Mani Tripathi

Volume-Preserving Mean Curvature Flow for Tubes in Rank One Symmetric Spaces of Non-compact Type

Abstract

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a closed geodesic ball in an invariant submanifold in a rank one symmetric space of non-compact type, where we impose some boundary condition to the flow and the invariancy of the submanifold means the total geodesicness in the case where the ambient symmetric space is a (real) hyperbolic space. Next, we prove that the tubeness is preserved along the flow in the case where the radius function of the initial tube is radial with respect to the center of the closed geodesic ball. Furthermore, in this case, we prove that the flow reaches to the invariant submanifold or it exists in infinite time and converges to a tube of constant mean curvature over the closed geodesic ball in the \(C^{\infty }\)-topology in infinite time.

Naoyuki Koike

A Duality Between Compact Symmetric Triads and Semisimple Pseudo-Riemannian Symmetric Pairs with Applications to Geometry of Hermann Type Actions

Abstract

This is a survey paper of not-yet-published papers listed in the reference as [1‐3]. We introduce the notion of a duality between commutative compact symmetric triads and semisimple pseudo-Riemannian symmetric pairs, which is a generalization of the duality between compact/noncompact Riemannian symmetric pairs. As its application, we give an alternative proof for Berger’s classification of semisimple pseudo-Riemannian symmetric pairs from the viewpoint of compact symmetric triads. More precisely, we give an explicit description of a one-to-one correspondence between commutative compact symmetric triads and semisimple pseudo-Riemannian symmetric pairs by using the theory of symmetric triads introduced by the second author. We also study the action of a symmetric subgroup of G on a pseudo-Riemannian symmetric space G / H, which is called a Hermann type action. For more details, see [1‐3].

Kurando Baba, Osamu Ikawa, Atsumu Sasaki

Transversally Complex Submanifolds of a Quaternion Projective Space

Abstract

We study a kind of complex submanifolds in a quaternion projective space \(\mathbb {H}P^n\), which we call transversally complex submanifolds, from the viewpoint of quaternionic differential geometry. There are several examples of transversally complex immersions of Hermitian symmetric spaces. For a transversally complex immersion \(f:M\rightarrow \mathbb {H}P^n \), a key notion is a Gauss map associated with f, which is a map \(S:M \rightarrow \mathrm{End}(\mathbb {H}^{n+1})\) with \(S^2 = -\mathrm{id}\). Our theory is an attempt of a generalization of the theory “Conformal geometry of surfaces in \(S^4\) and quaternions” by Burstall, Ferus, Leschke, Pedit, and Pinkall [4].

Kazumi Tsukada

On Floer Homology of the Gauss Images of Isoparametric Hypersurfaces

Abstract

The Gauss images of isoparametric hypersurfaces in the unit standard sphere provide compact minimal (thus monotone) Lagrangian submanifolds embedded in complex hyperquadrics. Recently we used the Floer homology and the lifted Floer homology for monotone Lagrangian submanifolds in order to study their Hamiltonian non-displaceability in our recent joint paper with Hiroshi Iriyeh, Hui Ma and Reiko Miyaoka. In this note we will explain the spectral sequences for the Floer homology and the lifted Floer homology of monotone Lagrangian submanifolds and their applications to the Gauss images of isoparametric hypersurfaces. They are the main technical part in our joint work. Moreover we will suggest some related open problems for further research.

Yoshihiro Ohnita

On the Pointwise Slant Submanifolds

Abstract

In this survey paper, we consider several kinds of submanifolds in Riemannian manifolds, which are obtained by many authors. (i.e., slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, pointwise semi-slant submanifolds, pointwise almost h-slant submanifolds, pointwise almost h-semi-slant submanifolds, etc.) And we deal with some results, which are obtained by many authors at this area. Finally, we give some open problems at this area.

Kwang-Soon Park

Riemannian Hilbert Manifolds

Abstract

In this article we collect results obtained by the authors jointly with other authors and we discuss old and new ideas. In particular we discuss singularities of the exponential map, completeness and homogeneity for Riemannian Hilbert quotient manifolds. We also extend a Theorem due to Nomizu and Ozeki to infinite dimensional Riemannian Hilbert manifolds.

Leonardo Biliotti, Francesco Mercuri

Real Hypersurfaces in Hermitian Symmetric Space of Rank Two with Killing Shape Operator

Abstract

We have considered a new notion of the shape operator A satisfies Killing tensor type for real hypersurfaces M in complex Grassmannians of rank two. With this notion we prove the non-existence of real hypersurfaces M in complex Grassmannians of rank two.

Ji-Eun Jang, Young Jin Suh, Changhwa Woo

The Chern-Moser-Tanaka Invariant on Pseudo-Hermitian Almost CR Manifolds

Abstract

We study on the Chern-Moser-Tanaka invariant (Chern, Acta Math 133:219–271, 1974, [5], Tanaka, Japan J Math 12:131–190, 1976, [14]) of pseudo-conformal transformations on pseudo-Hermitian almost CR manifolds.

Jong Taek Cho

Bott Periodicity, Submanifolds, and Vector Bundles

Abstract

We sketch a geometric proof of the classical theorem of Atiyah, Bott, and Shapiro [3] which relates Clifford modules to vector bundles over spheres. Every module of the Clifford algebra \(Cl_k\) defines a particular vector bundle over \(\mathord {\mathbb S}^{k+1}\), a generalized Hopf bundle, and the theorem asserts that this correspondence between \(Cl_k\)-modules and stable vector bundles over \(\mathord {\mathbb S}^{k+1}\) is an isomorphism modulo \(Cl_{k+1}\)-modules. We prove this theorem directly, based on explicit deformations as in Milnor’s book on Morse theory [8], and without referring to the Bott periodicity theorem as in [3].

Jost Eschenburg, Bernhard Hanke

The Solvable Models of Noncompact Real Two-Plane Grassmannians and Some Applications

Abstract

Every Riemannian symmetric space of noncompact type is isometric to some solvable Lie group equipped with a left-invariant Riemannian metric. The corresponding metric solvable Lie algebra is called the solvable model of the symmetric space. In this paper, we give explicit descriptions of the solvable models of noncompact real two-plane Grassmannians, and mention some applications to submanifold geometry, contact geometry, and geometry of left-invariant metrics.

Jong Taek Cho, Takahiro Hashinaga, Akira Kubo, Yuichiro Taketomi, Hiroshi Tamaru

Biharmonic Homogeneous Submanifolds in Compact Symmetric Spaces

Abstract

This paper is a survey of our recent works on biharmonic homogeneous submanifolds in compact symmetric spaces (Biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups (in preparation), Biharmonic homogeneous hypersurfaces in compact symmetric spaces. Differ Geom Appl 43, 155–179 (2015)) [12, 13]. We give a necessary and sufficient condition for an isometric immersion whose tension field is parallel to be biharmonic. By this criterion, we study biharmonic orbits of commutative Hermann actions in compact symmetric spaces, and give some classifications.

Shinji Ohno, Takashi Sakai, Hajime Urakawa

Recent Results on Real Hypersurfaces in Complex Quadrics

Abstract

In this survey article, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 2. Second, by using the isometric Reeb flow, we give a complete classification for hypersurfaces M in complex two-plane Grassmannians \(G_2({\mathbb C}^{m+2})=SU_{2+m}/S(U_{2}U_{m})\), complex hyperbolic two-plane Grassmannians \(G_{2}^{*}({\mathbb C}^{m+2})=SU_{2,m}/S(U_{2}U_{m})\), complex quadric \(Q^m={ SO}_{m+2}/SO_{m}SO_{2}\) and its dual \(Q^{m *}= SO_{m,2}^{o}/SO_{m}SO_{2}\). As a third, we introduce the classifications of contact hypersurfaces with constant mean curvature in the complex quadric \(Q^m\) and its noncompact dual \(Q^{m *}\) for \(m \ge 3\). Finally we want to mention some classifications of real hypersurfaces in the complex quadrics \(Q^m\) with Ricci parallel, harmonic curvature, parallel normal Jacobi, pseudo-Einstein, pseudo-anti commuting Ricci tensor and Ricci soliton etc.

Young Jin Suh

Backmatter