Taut submanifold

In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every [math]\displaystyle{ q\in M }[/math] the distance function

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

is a perfect Morse function.

If N is not compact, one needs to consider the restriction of the [math]\displaystyle{ L_q }[/math] to any of their sublevel sets.

References

• Hazewinkel, Michiel, ed. (2001), "Tight and taut immersions", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page