In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every q ∈ M {\displaystyle q\in M} the distance function
L q : N → R , L q ( x ) = dist ( x , q ) 2 {\displaystyle 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 {\displaystyle L_{q}} to any of their sublevel sets.
References Kuiper, N.H. (2001) [1994], "Tight and taut immersions", Encyclopedia of Mathematics, EMS Press