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.