On almost complex structures on tangent bundles

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(M,J,g)$ be a K\&quot;ahler--Norden manifold. Using the notions of the horizontal and vertical lifts, a class of almost complex structures $\widetilde J$ is defined on the tangent bundle $T\!M$, and necessary and sufficient conditions for such a structure to be integrable (complex) are described. Next, a class of pseudo-Riemannian metrics $\widetilde g$ of Norden type is defined on $T\!M$, for which $\widetilde J$ is an antiisometry. Thus, the pair $(\widetilde J,\widetilde g)$ becomes an almost complex structure with Norden metric on $T\!M$. It is checked whether the structure $(\widetilde J,\widetilde g)$ is K\&quot;ahler--Norden itself.     

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection on an almost Hermitian manifold M an almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h ¹ and h ² are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold.     

Para-CR Structures of Codimension 2 on Tangent Bundles in Riemann–Finsler Geometry

We determine a 2-codimensional para-CR structure on the slit tangent bundle T0 M of a Finsler manifold（M,F） by imposing a condition regarding the almost paracomplex structure P associated to F when restricted to the structural distribution of a framed para-f-structure.This condition is satisfied when（M,F） is of scalar flag curvature（particularly constant） or if the Riemannian manifold（M,g） is of constant curvature.     

Adjacent and tangent regularity

In this paper we define regularity for general adjacency and tangency concepts and we study it in some particular cases. Most of the convex but sophisticated approximations used in optimization theory can be disconvexified using the regularity notion and their adjacent or tangent structure can be perceived looking at them under the regularity angle.     

On regular tangent cones

The paper contains a sufficient condition for an intersection of regular tangent cones to be a tangent cone. Regular tangent cones and tents for sets given by locally Lipschitz functions are constructed. The cones are described in terms of generalized K-derivatives.