Classically, tightness is defined in terms of the total absolute curvature integral: a mapping of a surface into space is called tight if it has minimal total absolute curvature. [More]

This definition is valid only for smooth surfaces without boundary mapped into three-space. The modern definition has a more algebraic flavor:

Definition: A map f :M^m -> R^n is k-tight if, for all directions z and heights c, the map { p in M | z . f (p) <= c } -> M induces a monomorphism in the i-th Cech homology for each i from 0 to k.

Note that this definition is valid for manifolds of arbitrary dimension, both smooth an polyhedral, with or without boundary, in spaces of any dimension.

See also:

[More] Tightness and its consequences

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8/12/94 dpvc@geom.umn.edu -- The Geometry Center