Taut submanifold

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) [1994], "Tight and taut immersions", Encyclopedia of Mathematics, EMS Press