  In this paper we focus on algebraic aspects of contractions of Lie and
Leibniz algebras. The rigidity of algebras plays an important role in the study
of their varieties. The rigid algebras generate the irreducible components of
this variety. We deal with Leibniz algebras which are generalizations of Lie
algebras. In Lie algebras case, there are different kind of rigidities
(rigidity, absolutely rigidity, geometric rigidity and e.c.t.). We explore the
relations of these rigidities with Leibniz algebra rigidity. Necessary
conditions for a Lie algebra to be Leibniz rigid are discussed.
