Almost Analytic Kähler Forms with Respect to a Quadratic Endomorphism with Applications in Riemann-Finsler Geometry

Abstract

The almost analyticity with respect to a quadratic endomorphism T is introduced in an algebraic setting concerning a commutative and associative algebra $\mathcal {A}$. Two main properties are proved: the first concerns the simultaneous closedness for an almost analytic 1-form $\omega$ and $T\omega$ while the second regards the vanishing of the interior product of such a form with the Nijenhuis tensor of T. Also, we introduce an extension of the Frölicher–Nijenhuis formalism to this framework as well as a hermitian type property. When $\mathcal {A}$ is the algebra of smooth functions on a given (even dimensional) manifold we recover the classical notion of almost analytic 1-form. We study this analyticity and the hermitian type property for the Cartan 1-form of a Riemann-Finsler geometry. Also, we study the almost analytic functions on the tangent bundle of a Riemann-Finsler geometry with respect to the associated almost para-complex and almost complex structure of this geometry. We introduce two new types of Hessian and respectively Laplacian corresponding to these structures. Two types of gradient Ricci solitons are introduced in the tangent bundle.