Taut submanifold

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

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

is a perfect Morse function.^{[citation needed]}

If N is not compact, one needs to consider the restriction of the ${\displaystyle L_{q}}$ to any of their sublevel sets.