Naoyuki Koike Laboratory
　

Reserch Thema :

　
Differential Geometry and

　
Geometric Analysis

ORCID　KAKEN

Welcome to Naoyuki Koike Laboratory

Members of Koike laboratory study Differential Geometry which closely related to Einstein's General Relativity.
Mainly we study Submanifold Theory in Symmetric Spaces.

Main two methods of the research are as follows:

1.　Research of submanifolds in symmetric spaces of compact type
　　 〜　by reducing research of submanifolds in an infinite dimensional
　　　　　linear space obtained by canceling holonom of the symmetric space〜
We reasearch submanifolds in a symmetric space
of compact type by reducing to reasearch of submanifolds in an infinite dimensional Hilbert space through a Riemannian submersion of the Hilbert space onto the symmetric space. This reducement of research is valid because symmetric spaces of compact type has holonom but the Hilbert space has no holonom.

2.　Research of submanifolds in symmetric spaces of noncompact type
　　〜 by reducing to research of submanifolds in an infinite dimensional
　　　　complex linear space obtained by canceling holonom of the complexi-
　　　　fication of the symmetric space〜
We reasearch submanifolds in a symmetric space of compact type by reducing to reasearch of submanifolds in an infinite dimensional anti-Kaehlerian space through a Riemannian submersion of the anti-Kaehlerian space onto the complexification of the symmetric space. This reducement of research is valid in the following two aspects:
(1) A symmetric space of noncompact type has holonom but the infinite dimensional anti-Kaehlerian space has no holonom.
(2)　Some geometrical structures (of the submanifold) vanishing beyond the ideal boundary of a symmetric space of non-compact type appear in the complexified space.

3. 　Research of the mean curvature flow and the inverse mean curvature flow in 　　　symmetric spaces
We reaserch how a given submanifold collapses (or blows up) along the mean curvature flow and the invesrse mean curvature flow. Also, we research whether a given submanifold converges to a good submanifold along the suitably rescaled mean curvature flow and the suitably rescaled inverse mean curvature flow.

On 3 April, 2025, the following book plans to be published:
　　　 　 Signspot of Topological space
　　 　- From Foundation to Topological Invariants -
　　　　　　　　(Kyoritu Shuppan Co., Ltd.)
　　　　　　　　　　　　　　　　　　　　　　　　(Jump to sight)

On 12 September, 2022, the following book plans to be published:
　　　 Vector Analysis and Differential Geometry
　　　　Enlightenned by Integral Formulas
　　 - From Stokes's Theorem to the Variational Formulas -
　　　　　　　　(Kyoritu Shuppan Co., Ltd.)
　　　　　　　　　　　　　　　　　　　　　　　　(Jump to sight)

On 9 March, 2021, the following book plans to be published:
　　　 　 　　Submanifold Geometry
　　　　　Hidden in Theoretical Physics
　 - Connection with General Theory of Relativity and Gauge theory -
　　　　　　　　(Kyoritu Shuppan Co., Ltd.)
　　　　　　　　　　　　　　　　　　　　　　　　(Jump to sight)

On 10 April, 2019, the following book was published:
　　　 　　　　Mean Curvature flow
　　　　　- Time Evolution of Submanifolds -
　　　　　　　　(Kyoritu Shuppan Co., Ltd.)
　　　　　　　　　　　　　　　　　　　　　　　　(Jump to sight)

Photos of Cats and Bears

Announcement(April,2019-March,2020)

・　I plan to talk about ``Isoprametric submanifolds admitting a reflective
　focal submanifold in symmetric spaces of non-compact type''
　at Workshop on the isoparametric theory''
　Date：June 2-6
　Place：Beijing Normal University, China


・　Conference ``Submanifold Geometry and Lie Group Action 2018''
　was held.
　Date：September 3-4
　Place：Tokyo University of Science, Morito Memorial Hall
　See the following site in detail:
　http://www.rs.kagu.tus.ac.jp/~koike/sgla2018.html


・　I talked about ``Mean curvature flow for pinched submanifolds in rank
　one symmetric spaces''
　at The 64-th Geometry Symposium
　Date：August 29
　Place：Kanazawa University

・　Conference ``Geometric flows and related problems''
　plan to be held.
　Date：November 23-25
　Place：Tokyo Institute of Technology, Ookayama Campus
　See the following site in detail:
　http://www.rs.kagu.tus.ac.jp/~koike/gflow2017-e.html

・　The 62-th Geometry Symposium was held.
　Date：August 27-30
　Place：Tokyo University of Science, Kagurazaka Campus
　See the following site in detail:

http://www.rs.kagu.tus.ac.jp/~koike/Geometry-Symposium-2015.html

・　Conference ``Submanifold Geometry and Lie Group Action 2015''
　was held.
　Date：September 7-8
　Place：Tokyo University of Science, Morito Memorial Hall
　See the following site in detail:
　http://www.rs.kagu.tus.ac.jp/~koike/sgla2015.html

・Conference
　``Geometric Analysis in Geometry and Topology 2015''
　plans to be held.
　Date：November 9-12
　Place：Tokyo University of Science, Morito Memorial Hall
　See the following site in detail:
　http://www.rs.kagu.tus.ac.jp/~koike/gagt2015.html　

・Conference
　``Geometric flows and related problems'' plans to be held.
　Date：March 3-4, 2016
　Place：Tokyo Institute of Technology (Ookayama campus)
　See the following site in detail:
　http://www.rs.kagu.tus.ac.jp/~koike/gflow.html　

・I talked about ``the regularized mean curvature flow in the Hilbert space'' in
　　``2014 ICM Satellite Cinference on Real and Complex Submanifolds''.
　　Date : August, 10-12
　 Place：NIMS, Daejeon, Korea
See the following site about the detail:http://www.icm2014.org/en/program/satellite/satellites

・Conference ``Submanifold Geometry and Lie Group Action 2014''
　was held.
　Date：September 5-６
　Place：Tokyo University of Science, Morito Memorial Hall
　See the following site in detail:
　http://www.rs.kagu.tus.ac.jp/~koike/sgla2014-e.html

・Conference
``Geometric Analysis in Geometry and Topology 2014''
　was held.
　　Date：october 28-31
　 Place：Tokyo University of Science, Morito Memorial Hall
　See the follwong site in detail:
　http://www.rs.kagu.tus.ac.jp/~koike/gagt2014.html