On the total absolute curvature of closed curves in manifolds of negative curvature

1974 ◽  
Vol 210 (4) ◽  
pp. 313-319 ◽  
Author(s):  
Y�tar� Tsukamoto
Keyword(s):  
Negative Curvature ◽  
Closed Curves ◽  
Total Absolute Curvature ◽  
Absolute Curvature

scholarly journals Nonpositive curvature foliations on 3-manifolds with bounded total absolute curvature of leaves

2019 ◽  
Vol 11 (4) ◽  
pp. 1-17
Author(s):  
Dmytry Bolotov
Keyword(s):  
Negative Curvature ◽  
Positive Curvature ◽  
Nonpositive Curvature ◽  
Riemannian Metric ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Total Absolute Curvature ◽  
New Class ◽  
Absolute Curvature

In this paper we introduce a new class of foliations on Rie-mannian 3-manifolds, called B-foliations, generalizing the class of foliations of non-negative curvature. The leaves of B-foliations have bounded total absolute curvature in the induced Riemannian metric. We describe several topological and geometric properties of B-foliations and the structure of closed oriented 3-dimensional manifolds admitting B-foliations with non-positive curvature of leaves.

scholarly journals (n-2)-tightness and curvature of submanifolds with boundary

1978 ◽  
Vol 1 (4) ◽  
pp. 421-431 ◽  
Author(s):  
Wolfgang Kühnel
Keyword(s):  
Euclidean Space ◽  
Geometric Inequality ◽  
Total Absolute Curvature ◽  
Absolute Curvature

The purpose of this note is to establish a connection between the notion of(n−2)-tightness in the sense of N.H. Kuiper and T.F. Banchoff and the total absolute curvature of compact submanifolds-with-boundary of even dimension in Euclidean space. The argument used is a certain geometric inequality similar to that of S.S. Chern and R.K. Lashof where equality characterizes(n−2)-tightness.

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