2015-10-26

Geometry of Lagrangian submanifolds in complex hyperquadrics
Chairman Jiazu Zhou

The complex hyperquadric $Q_{n}({\mathbb C})$ is canonically identified with
the real Grassmann manifold $\widetilde{Gr}_{2}({\mathbb R}^{n+2})$
of all oriented $2$-dimensional vector subspaces of ${\mathbb R}^{n+2}$,
which is a compact Hermitian symmetric space of rank two (if $n\geq 2$).
In general it is an interesting problem to study submanifold geometry
in symmetric space of rank greater than one.
In this talk we shall discuss geometry of Lagrangian submanifolds in complex hyperquadrics:
We shall begin with elementary properties and examples of such Lagrangian submanifolds
and emphasis on the relationship with the hypersurface geometry in the standard unit sphere
such as the Gauss map construction of Lagrangian submanifolds.
This talk is the first one of my two lectures on this area
at this international workshop.

From Euclidean geometry to manifold theory and some curvature properties in Riemannian geometry

Chairman Jiazu Zhou

In the present talk we explain how the notion of manifolds come from Euclidean geometry. Next some basic properties of curvature tensors in Riemannian geometry have been discussed. In particular, 2-dimensional and 3-dimensional Riemannian space have been considered.

Isoparametric foliation and a problem of Besse

Chairman Yoshihiro Ohnita

This talk gives a survey on the recent progress in the study of Willmore, Einstein and some Einstein-like properties of the focal submanifolds of isoparametric hypersurfaces in spheres with $g = 4$ and $g = 6$ distinct principal curvatures.

Session 1

$(\eta, \eta_{a}, \theta)$-Einstein real hypersurfaces in complex two-plane Grassmannians

Chairman Makiko Sumi Tanaka

In this paper, we introduce the notion of $(\eta,\eta_a,\theta)$-Einstein real hypersurfaces in complex two-plane Grassmannians. We show that there does not exist any $(\eta,\eta_a,\theta)$-Einstein real hypersurface in complex two-plane Grassmannians such that $\xi$ is tangent to $\mathfrak D$. Some examples of $(\eta_a,\theta)$-Einstein real hypersurfaces are given.

Session 2

Quantum and Floer type cohomologies on the almost contact metric manifolds with closed fundamental forms

Chairman Nobuhiro Innami

To construct the quantum type cohomologies, we study pseudo-coholomorphic curves, moduli spaces of the curves representing $2$-dimensional homology classes, Gromov-Witten type invariants, and quantum type product on cohomology groups. For Floer type cohomologies on the manifolds, we study a symplectic type action funtional on the universal covering space of the space of contractible loops. The critical points of the funtional and the moduli space of gradient flow lines joining critical points induce a cochain complex and produce a Floer type cohomology. We show that the quantum type cohomology and the Floer type cohomology on the manifolds are isomorphic.

Session 1

On concircular geometry in the Riemannian manifold

Chairman Makiko Sumi Tanaka

The concircular transformation of Riemannian manifolds was introduced by K.Yano in 1940, which is characterized by a conformal transformation preserving geodesic circle. The notion of concircular vector fields was introduced by A.Falkow in 1939 and developed the concircular geometry and many geometers obtained interesting results. Moreover a Ricci soliton on a Riemannian manifold is said to have concircular potential field if its potential field is a concircular vector field.
In this talk, we summarize for recent results for the concircular geometry and Ricci solitons. Moreover we consider and discuss about concircular geometry and Ricci soliton related problems in the Riemannian manifold.

Session 2

Star-Ricci flat real hypersurfaces in complex space forms

Chairman Nobuhiro Innami

The study of real hypersurfaces in complex space forms has been an
active field of study over the past decade.
Recently, Kaimakamis and Panagiotidou classified real hypersurfaces with
parallel star-Ricci tensor in complex projective plane $P_{2}(\mathbf{C})$ and
complex hyperbolic plane $H_{2}(\mathbf{C})$. We can find the relations with their
result and flat star-Ricci tensor.
In this paper, we classified real hypersurfaces of
complex space forms $P_{n}(\mathbf{C})$ and $H_{n}(\mathbf{C})$ with flat star-Ricci tensor.

Antipodal structure of the intersection of real flag manifolds in a complex flag manifold II

Chairman Makiko Sumi Tanaka

Tanaka and Tasaki studied the antipodal structure of the intersection of two real forms in Hermitian symmetric spaces of compact type. An orbit of the adjoint representation of a compact connected Lie group $G$ admits a $G$-invariant K ̈ahler structure, and called a complex flag manifold. Furthermore, any simply-connected compact homogeneous K ̈ahler manifold is a complex flag manifold. Using $k$-symmetric structures, we can define (generalized) antipodal sets of a complex flag manifold. An orbit of the linear isotropy representation of the compact symmetric space $G/K$ is called a real flag manifold, and is embedded in a complex flag manifold as a real form. In this talk, we will give a necessary and sufficient condition for two real flag manifolds, which are not necessarily congruent with each other, in a complex flag manifold to intersect transversally in terms of symmetric triads. Moreover we will show that the intersection is an orbit of a certain Weyl group and an antipodal set, if the intersection is discrete.

On Bonnesen-style symmetric mixed inequality

Chairman Young Jin Suh

The symmetric mixed isoperimetric deficit $\Delta_2(K_0,K_1)$ of domains $K_0$ and $K_1$ in the Euclidean plane $R^2$ is defined in this paper. The symmetric mixed isoperimetric inequality and some new Bonnesen-style symmetric mixed inequalities are obtained via the known kinematic formulae of Poincar\'e and Blaschke in integral geometry.

2015-10-27

Certain generalizations of Einstein manifolds with applications to relativity
Chariman Young Jin Suh

We study quasi-Einstein and generalized quasi-Einstein manifolds which are the
natural generalizations of Einstein manifolds. First we consider quasi-Einstein
manifolds. We obtain some results regarding quasi-Einstein manifolds and give
some examples of such manifolds. We also study pseudo Ricci symmetric and special type of quasi-Einstein manifolds and give examples of such manifolds which justify some theorems. Next, we study generalized quasi-Einstein manifolds. Finally,
we give some applications of quasi-Einstein manifolds in the theory of relativity.

Some new aspects of the Bernstein theorem I

Chariman Young Jin Suh

There are various generalizations of the well-known Bernstein theorem for minimal graphs in $\mathbb R^{3}$ over the past decades. We will review some recent developments for minimal parametric hypersurfaces in Euclidean space, higher codimensional generalizations for Moser's weak version of Bernstein's theorem (so-called Lawson-Osserman problem), as well as the Bernstein type theorems for minimal hypersurfaces in general Riemannian manifolds with non-negative Ricci curvature.

Geometry of the Gauss images of isoparametric hypersurfaces

Chairman Yasuo Matsushita

Hypersurfaces with constant principal curvatures in the standard sphere are so-called isoparametric hypersurafaces. Through the Gauss map, the isoparametric hypersurafaces in the standard unit sphere provide a nice class of Lagrangian submanifolds in complex hyperquadrics. We know that the Gauss image, i.e. the image of the Gauss map, of each isoparametric hypersuraface in the standard unit sphere is a compact minimal Lagrangian submanifold embedded in the complex hyperquadric. We studied geometric and topological properties of so obtained Lagrangian submanifolds such as the minimal Maslov number, the classification problem of homogeneous Lagrangian submanifolds and the Hamiltonian stability problem. Moreover, we shall explain recent results on Hamiltonian non-displaceablity of the Gauss images of isoparametric hypersurafaces, which the Lagrangian Floer theory can be applied to. This lecture is based on my joint works with Hui Ma, Hiroshi Iriyeh and Reiko Miyaoka.
This talk is the second one of my two lectures at this international workshop.

Session 1

Symmetries on real hypersurfaces in Kahlerian manifolds

Chairman Takashi Sakai

A real hypersurface of K ̈ahlerian manifold admits the two fundamental structures
induced from the K ̈ahlerian structure of ambient space. One is Riemannian struc-
ture and the other is CR structure. In this context, we may consider two natural
affine connections on real hypersurfaces of K ̈ahlerian manifold: the Levi-Civita
connection and the generalized Tanaka-Webster connection. In this talk, we study
several symmetries using these two connections respectively.

Session 2

Geodesics in a Finsler torus of revolution

Chairman Uday Chand De

Let $(M, F)$ denote a Finsler surface. Namely, $M$ is a differentiable $2$-manifold with fundamental function $F:TM\to\mathbb{R}$ such that $F$ is smooth and positive on $TM \smallsetminus \{ 0 \}$, $F(x, t\dot x)=tF(x, \dot x)$, $t > 0$, for all $\dot x \in T_xM$ and $F(x, \dot x)$ is strictly convex for $\dot x \not= 0$. Here $TM$ denotes the tangent bundle of $M$.
The intrinsic distance $d$ on $M$ is defined by $d(p,q):=\inf\{ L_F(c)\, |\, c \in \Omega (p, q) \}$,
where $\Omega (p,q)$ denotes the set of all piecewise smooth curves from $p$ to $q$ and $L_F(c)$ is the length of a curve $c:[0,1]\to M$.
An extremal of the variation problem of lengths of curves $c \in \Omega (p, q)$ is called a geodesic in $(M, F)$.
Let $T(p, q)$ denote a minimizing geodesic segment from $p$ to $q$.
If $F$ is not absolutely homogeneous, then the distance $d$ is not symmetric and $T(p,q) \not= T(q,p)$, in general.
When $M$ is orientable, an isometry which preserves the orientation of $M$ is called a {\em motion} on $(M, F)$. We say that $\varphi_t \, : \, M \rightarrow M$, $t \in (-\infty, \infty)$, is a {\em one-parameter group of motions} on $(M, F)$ if $\varphi_t$ satisfies $F(\varphi_t(x), \varphi_{t*}(\dot x))=F(x, \dot x)$ for all $x \in M$, $\dot x \in T_xM$ and all $t \in (-\infty , \infty)$.
We study the global behavior of geodesics in a complete Finsler surface with one-parameter group of motions. In particular, we determine what types of geodesics there are in a Finsler 2-torus with revolution with non-symmetric distance.

Session 1

Real hypersurfaces in a 2-dimensional complex space form with transversal Killing tensor fields

Chairman Takashi Sakai

Let $M$ be a real hypersuface in a non-flat complex space form $M^2(c)$. The shape operator $A$ is called transversal Killing tensor field if it satisfies $(\nabla_X A)X=0$ for any vector field $X\perp \xi$, where $\xi$ is a structure vector field. We prove that $A$ is transversal Killing tensor field if and only if $M$ is a totally umbilical real hypersurface.

Session 2

Real toric manifolds, links, and integral cohomology groups

Chairman Uday Chand De

In view of various results and the nature of their constructions for real toric objects, it is an intriguing question to answer how much amount of torsion can be contained in their integral cohomology groups. In this talk, I want to explain how to explicitly construct certain examples of real toric objects, called the real quasitoric manifolds, which shows the richness of their torsion. To do so, we first give some necessary background for the notions such as links, quasitoric manifolds, and their real versions. We then describe some important steps for the construction, relatively in detail.

Session 1

Classification of generalized Sasakian space forms

Chairman Takashi Sakai

In the present paper we review the advancement of the theory of generalized Sasakian spaceforms. We give a local characterization of generalized Sasakian space forms of dimension greater than 5. We show that it is of constant sectional curvature, or an $(\alpha, \beta)$ trans-Sasakian manifold, or a canal hypersurface in a Euclidean space or Minkowski space, or a certain type of twisted product of $\mathbb R$ and a flat almost Hermitian manifold.

Session 2

The Olivier Rey's inequality on the Heisenberg group

Chairman Uday Chand De

We will study CR analogue of the Olivier Rey's inequality on the Heisenberg group. In confromal setting this inequality is used to prove the existence of the solution to the linearized Yamabe equation. This inequality shows that the energy functional for perturbed Yamabe equation is bounded below, if the perturbation is small enough.
In this article, we identify the Heisenberg group and the standard sphere via the Cayley transformation, and analyze the eigenvalues of the sub-Laplacian $\Delta_b$ on $S^{2n + 1}$.

Neutral geometry in 4-dimension and counter examples to Goldberg conjecture constructed on Walker 6-manifolds

Chairman Byung Hak Kim

Our talk covers two topics, the first being a survey of the existence problem for an indefinite metric of neutral signature $(+2,-2)$ on a compact orientable 4-manidold, while the second presents a new counterexample to the Goldberg conjecture provided by an indefinite metric of signature $(+4,-2)$ on a compact six-dimensional manifold (a six-torus).

2015-10-28

Some new aspects of the Bernstein theorem II
Chairman Yong Seung Cho

There are various generalizations of the well-known Bernstein theorem for minimal graphs in $\mathbb R^{3}$ over the past decades. We will review some recent developments for minimal parametric hypersurfaces in Euclidean space, higher codimensional generalizations for Moser's weak version of Bernstein's theorem (so-called Lawson-Osserman problem), as well as the Bernstein type theorems for minimal hypersurfaces in general Riemannian manifolds with non-negative Ricci curvature.

Session 1

Constant f-mean curvature surfaces in a smooth metric measure space

Chairman Tatsuyoshi Hamada

In this talk, we introduce surface theory in a smooth metric measure space and
give some rigidity results of constant f-mean curvature surfaces in a smooth metric
measure space.

Session 2

On generalized Sasakain space-forms

Chairman Yuan Long Xin

The object of the present paper is to study some curvature properties of generalized Sasakian space-forms. First we consider Ricci pseudosymmetric generalized Sasakian space-forms. We also study Ricci generalized pseudosymmetric and Weyl semisymmetric generalized Sasakian space-forms. Next, we consider $\xi$-conformally flat, $\phi$-conformally flat, $\phi$-Weyl semisymmetric, $\phi$-projectively semisymmetric generalized Sasakian space-forms. As a consequence of the results we deduce some important corollaries. Finally, illustrative examples are given.

Session 1

Characterizations for real hypersurfaces in complex two-plane Grassmannians related to normal Jacobi operator

Chairman Tatsuyoshi Hamada

In this talk, we introduce a new notion of Reeb parallel normal
Jacobi operator for homogeneous real hypersurfaces in complex two-plane Grassmannians which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. And we give a complete classification for real hypersurfaces of Type~$(A)$ in complex two-plane Grassmannians $G_{2}(\mathbb C^{m+2})$, that is, a tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in $G_{2}(\mathbb C^{m+2})$ with Reeb parallel normal Jacobi operator.

Session 2

Some hypersurfaces in GTW Reeb Lie derivative structure Jacobi operator in complex two-plane Grassmannians

Chairman Yuan Long Xin

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. And using the method of simultaneous diagonalization, we give 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 $G_2({\mathbb C}^{m+2})$.

Session 1

On a Riemannian submanifold whose slice representation has no nonzero fixed point

Chairman Tee-How Loo

In this talk, we introduce a new class of Riemannian submanifolds which we call
arid submanifolds. We say a Riemannian submanifold is arid if no nonzero normal
vectors are invariant under the full slice representation. Arid submanifolds are a
generalization of the notion of weakly reflective submanifolds introduced by Ikawa,
Sakai and Tasaki. On the other hand, any arid submanifolds are minimal. We
also introduce an application of the idea of arid submanifolds to the study of left-
invariant metrics on a Lie group. We have obtained a sufficient condition for an
arbitrary Riemannian Lie group to be a Ricci soliton.

Session 2

Classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with Ricci tensors

Chairman Mayuko Kon

We have studied classifying problem of immersed submanifolds in Hermitian sym-
metric spaces. Typically in this paper, we consider a new notion of Reeb parallel
Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two-plane
Grassmannians which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. By using a common basis of simultaneously diagonalizable
finite dimensional operators, we give a complete classification for real hypersurfaces
in complex hyperbolic two-plane Grassmannians with the given condition.

Session 1

Study on Jacobi operators in complex two-plane Grassmannians

Chairman Tee-How Loo

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. And using the method of simultaneous diagonalization, we give 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 $G_2({\mathbb C}^{m+2})$.
References
[1] J. Berndt, Riemannian geometry of complex two-plane Grassmannian, Rend. Sem. Mat.
Univ. Politec. Torino 55 (1997), 19–83.
[2] I. Jeong, H. J. Kim and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians
with parallel normal Jacobi operator, Publ. Math. Debrecen 76 (2010), 203–218.
[3] H. Lee, J. D. P ́
erez and Y.J. Suh, On the structure Jacobi operator of a real hypersurface in
complex projective space, Monatsh. Math. 158 (2009), no. 2, 187–194.

Session 2

A generalized invariance of the structure Jacobi operator for real hypersurfaces in compact complex Grassmanians of rank 2

Chairman Mayuko Kon

In this talk, first we introduce a new invariance of the derivative $L_{\xi_{\mu}}-\nabla_{\xi_{\mu}}$ for the structure Jacobi operator $R_{\xi}:=R(\cdot, \xi)\xi$ on real hypersurfaces in complex two-plane Grassmannians $G_{2}(\mathbb{C}^{m+2})$, where $L_{\xi_{\mu}}$ ($\nabla_{\xi_{\mu}}$, resp.) denotes the Lie (covariant, resp.) derivative along the direction of $\xi_{\mu} \in \mathcal Q^{\bot}$ and $\mathcal Q^{\bot} \subset TM$ a 3-dimensional distribution spanned by $\xi_{\mu}=-J_{\mu}N$ ($\mu=1,2,3$). By using this notion, we give a new characterization of Hopf hypersurfaces in $G_{2}(\mathbb{C}^{m+2})$, $m \geq 3$.

Neutral geometry in 4-dimension, Walker 4-manifolds and the Spinor approach

Chairman Young Jin Suh

This talk is a survey on neutral geometry in 4-dimension. Among pseudo-Riemannian manifolds of various indefinite metric signatures, such a neutral metric in 4-dimension exhibits many specific properties. These interesting aspects and significant structures will be shown. Especially, Walker 4-manifolds and the spinor approach to the neutral 4-geometry will be also included.

Isometries of extrinsic symmetric spaces

Chairman Young Jin Suh

We show that every isometry of an extrinsic symmetric space extends to an isometry of its ambient euclidean space. As a consequence, any isometry of a real form of a Hermitian symemtric space extends to a holomorphic isometry of the ambient Hermitian symmetric space. Moreover, every fixed point component of an isometry of a symmetric $R$-space is a symmetric $R$-space itself. This article is based on the joint work with J. -H. Eschenburg and P. Quast.