 
Summary: TAUTNESS IS INVARIANT UNDER LIE SPHERE
TRANSFORMATIONS
J.C. ´ALVAREZ PAIVA
Abstract. In 1987, Cecil and Chern showed that tautness is invariant under
Lie sphere transformations. This note presents a very simple proof of this result
and a manifestly invariant definition of tautness for Legendrian submanifolds
in the space of cooriented contact elements of the sphere.
1. Introduction
An embedded compact submanifold M Sn
is said to be taut if for almost
every point x Sn
, the distancesquared function m d(x, m)2
has as many
critical points as the sum of the Z2Betti numbers of M. An equivalent, more
popular, definition (see [3], pp. 113) is that for every closed (geodesic) ball B Sn
the induced homomorphism H(B M) H(M) in Chech homology with Z2
coefficients is injective.
Both definitions are useful: the first trivially implies that the parallel surfaces
to a taut submanifold are also taut, while the second trivially implies that tautness
is invariant under conformal transformations. In this paper we give yet another
