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In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every \( q\in M\, \) the distance function

\( L_q:N\to\mathbf R,\qquad L_q(x) = \operatorname{dist}(x,q)^2\, \)

is a perfect Morse function.

If N is not compact, one needs to consider the restriction of the L_q to any of their sublevel sets.
References

Kuiper, N.H. (2001), "Tight and taut immersions", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Mathematics Encyclopedia

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