We express the dual forms of squares of Nijenhuis tensor in terms of the second order component derivatives of the exterior derivative on differential forms and give new vanishing results for the squares of Nijenhuis tensor.
We introduce a notion of compatibility between (almost) Dirac structures and (1,1)-tensor fields extending that of Poisson-Nijenhuis structures. We study several properties of the Dirac-Nijenhuis structures thus obtained, including their connection w
ith holomorphic Dirac structures, the geometry of their leaves and quotients, as well as the presence of hierarchies. We also consider their integration to Lie groupoids, which includes the integration of holomorphic Dirac structures as a special case.
The space of vector-valued forms on any manifold is a graded Lie algebra with respect to the Frolicher-Nijenhuis bracket. In this paper we consider multiplicative vector-valued forms on Lie groupoids and show that they naturally form a graded Lie sub
algebra. Along the way, we discuss various examples and different characterizations of multiplicative vector-valued forms.
The conullity of a curvature tensor is the codimension of its kernel. We consider the cases of conullity two in any dimension and conullity three in dimension four. We show that these conditions are compatible with non-negative sectional curvature on
ly if either the manifold is diffeomorphic to $mathbb{R}^n$ or the universal cover is an isometric product with a Euclidean factor. Moreover, we show that finite volume manifolds with conullity 3 are locally products.
In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Bach tensor and Weyl tensor implies the harmonicity of the metric, and we p
resent examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. On the other hand, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.