An introduction to infinite-dimensional differential geometry
著者
書誌事項
An introduction to infinite-dimensional differential geometry
(Cambridge studies in advanced mathematics, 202)
Cambridge University Press, 2023
- : hardback
大学図書館所蔵 全23件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
注記
Includes bibliographical references (p. 256-263) and indexes
内容説明・目次
内容説明
Introducing foundational concepts in infinite-dimensional differential geometry beyond Banach manifolds, this text is based on Bastiani calculus. It focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry, exploring their connections to manifolds of (smooth) mappings. Topics covered include diffeomorphism groups, loop groups and Riemannian metrics for shape analysis. Numerous examples highlight both surprising connections between finite- and infinite-dimensional geometry, and challenges occurring solely in infinite dimensions. The geometric techniques developed are then showcased in modern applications of geometry such as geometric hydrodynamics, higher geometry in the guise of Lie groupoids, and rough path theory. With plentiful exercises, some with solutions, and worked examples, this will be indispensable for graduate students and researchers working at the intersection of functional analysis, non-linear differential equations and differential geometry. This title is also available as Open Access on Cambridge Core.
目次
- 1. Calculus in locally convex spaces
- 2. Spaces and manifolds of smooth maps
- 3. Lifting geometry to mapping spaces I: Lie groups
- 4. Lifting geometry to mapping spaces II: (weak) Riemannian metrics
- 5. Weak Riemannian metrics with applications in shape analysis
- 6. Connecting finite-dimensional, infinite-dimensional and higher geometry
- 7. Euler-Arnold theory: PDE via geometry
- 8. The geometry of rough paths
- A. A primer on topological vector spaces and locally convex spaces
- B. Basic ideas from topology
- C. Canonical manifold of mappings
- D. Vector fields and their Lie bracket
- E. Differential forms on infinite-dimensional manifolds
- F. Solutions to selected exercises
- References
- Index.
「Nielsen BookData」 より