Last update : September 27, 2016

# 第１５回 秋葉原微分幾何セミナー

 日程 ： ２０１６年３月２５日（金） 場所 ： 東京都千代田区外神田1-18-13 秋葉原ダイビル12階　首都大学東京 秋葉原サテライトキャンパス会議室ＤＥ 講演者 ： Jost Eschenburg 氏（Universität Augsburg） 講演題目 ： Extrinsic Symmetric Spaces

#### Program :

 13:30-14:30 Lecture 1. A geometric motivation: Parallel second fundamental form. Abstract: Any curve with constant nonzero curvature in euclidean plane is a circle. The subject of our talks is a generalization to arbitrary dimension and codimension. Curves are replaced by submanifolds, curvature by the second fundamental form. Submanifolds with constant (= parallel) second fundamental form turn out to be objects of fundamental importance in basic mathematics; they include the matrix groups O(n), U(n), Sp(n) and the Grassmannians (the set of all vector subspaces of a given dimension). From the geometric view point they still resemble the planar circle, being invariant under reflection along each of their normal spaces ("extrinsic symmetric"). The aim of the first talk is to indicate a common construction for all these objects: they form certain orbits of the rotation (isotropy) groups of other symmetric spaces (= Riemannian spaces with an isometric point reflection at every point). 14:50-15:50 Lecture 2. Intrinsic and extrinsic geometry. Abstract: Extrinsic symmetric spaces form a subfamily of symmetric spaces. What are their special features? Either they carry a parallel complex structure (Hermitian symmetric spaces) or they are real forms of such spaces. They are closely related to the so called center of other symmetric spaces. Their maximal tori are also extrinsic symmetric and therefore an orthogonal product of planar circles. All their isometries extend to isometries of the ambient space. Extrinsic symmetric spaces are embedded not only in euclidean space, but also in in other symmetric spaces where they are again extrinsic symmetric. In fact these are the only full extrinsic symmetric subspaces of symmetric spaces. 16:10-17:10 Lecture 3. The noncompact transformation group. Abstract: Extrinsic symmetric spaces are compact, but they allow a noncompact group of transformations extending the isometry group. E.g. for the round sphere this is the group of conformal transformation, for projective space it is the projective linear group. This noncompact group is characteristic for all extrinsic symmetric spaces. It is generated by the gradient flow of the height functions on the ambient space. Every compact symmetric space has a noncompact dual symmetric space; for extrinsic symmetric spaces this is equivariantly embedded as an open subset of the compact space where the isometry group of the dual becomes part of the noncompact group on the extrinsic symmetric space.

#### アクセス：

 ＪＲ 山手線・京浜東北線・総武中央線 「秋葉原駅」 徒歩約1分 つくばエクスプレス 「秋葉原駅」 徒歩約2分 東京メトロ日比谷線 「秋葉原駅」 徒歩約5分 東京メトロ日銀座線 「末広町駅」 徒歩約5分

#### 世話人：

 田崎 博之 （筑波大学数理物質） 田中 真紀子 （東京理科大学理工） 小野 肇 （埼玉大学理工） 酒井 高司 （首都大学東京理工）