  First we use a new approach to give a graded Lie algebra whose Maurer-Cartan
elements characterize pre-Lie algebra structures. Then using this graded Lie
bracket we define the notion of a Nijenhuis operator on a pre-Lie algebra which
generates a trivial deformation of this pre-Lie algebra. There are close
relationships between O-operators, Rota-Baxter operators and Nijenhuis
operators on a pre-Lie algebra. In particular, a Nijenhuis operator "connects"
two O-operators on a pre-Lie algebra whose any linear combination is still an
O-operator in certain sense and hence compatible L-dendriform algebras appear
naturally as the induced algebraic structures. For the case of the dual
representation of the regular representation of a pre-Lie algebra, there is a
geometric interpretation by introducing the notion of a
pseudo-Hessian-Nijenhuis structure which gives rise to a sequence of
pseudo-Hessian and pseudo-Hessian-Nijenhuis structures. Another application of
Nijenhuis operators on pre-Lie algebras in geometry is illustrated by
introducing the notion of a para-complex structure on a pre-Lie algebra and
then studying paracomplex quadratic pre-Lie algebras and paracomplex
pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of
Nijenhuis operators on pre-Lie algebras.
