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Total Curvature of Manifolds with Boundary in E n
Author(s) -
Kühnel Wolfgang
Publication year - 1977
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-15.1.173
Subject(s) - mathematics , normal bundle , boundary (topology) , manifold (fluid mechanics) , mathematical analysis , sectional curvature , curvature , pure mathematics , bundle , total curvature , mean curvature , scalar curvature , geometry , vector bundle , mechanical engineering , materials science , engineering , composite material
By analogy with the case of closed manifolds in E n we consider the (extrinsic) Lipschitz‐Killing curvature of an immersion of a compact manifold with boundary. For this purpose the unit normal bundle has to be replaced by the union of the unit normal bundle of the interior and the “outer” unit normal bundle of the boundary. We prove theorems of the Chern‐Lashof type ( cf . [4]) and other necessary conditions for the topological structure of a manifold admitting an immersion with small total absolute curvature, and the Gauss‐Bonnet theorem.