References Abbassi, M. T. K.; Sarih, M. - On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds , Differential Geom. Appl., 22 (2005), 19-47. Anastasiei, M. - Locally conformal Kaehler structures on tangent manifold of a space form , Libertas Math., 19 (1999), 71-76. Bejan, C. L.; Benyounes, M. - Kähler manifolds of quasi-constant holomorphic sectional curvature , J. Geom., 88 (2008), 1-14. Bejan, C. L.; Oproiu, V. - Tangent bundles of quasi-constant holomorphic sectional curvatures , Balkan J. Geom. Appl

References [1] M.T.K. Abbassi, G. Calvaruso, D. Perrone: Harmonic sections of tangent bundles equipped with Riemannian g-natural metrics. Q. J. Math. 62 (2) (2011) 259–288. [2] R. M. Aguilar: Isotropic almost complex structures on tangent bundles. Manuscripta Math. 90 (4) (1996) 429–436. [3] I. Biswas, J. Loftin, M. Stemmler: Flat bundles on affine manifolds. Arabian Journal of Mathematics 2 (2) (2013) 159–175. [4] J. Choi, A. P. Mullhaupt: Kählerian information geometry for signal processing. Entropy 17 (2015) 1581–1605. [5] R. M. Friswell, C. M. Wood

## Abstract

We define a class of metrics that extend the Sasaki metric of the tangent manifold of a Riemannian manifold. The new metrics are obtained by the transfer of the generalized (pseudo-)Riemannian metrics of the pullback bundle π−1(TM⊕T*M), where π : T M → M is the natural projection. We obtain the expression of the transferred metric and define a canonical metric connection with torsion. We calculate the torsion, curvature and Ricci curvature of this connection and give a few applications of the results. We also discuss the transfer of generalized complex and generalized Kähler structures from the pullback bundle to the tangent manifold.

## Riemannian Metrics on the Tangent Bundle of a Finsler Submanifold

Let IF^{n} = (*M, F*) be a Finsler submanifold of a Finsler manifold IF^{n+p} = (*M, F*). Then the induced non-linear connection *HTM*° and the canonical non-linear connection *GTM*° define two Riemannian metrics *G* and *G*
^{*} on *™*° = *™* \ {0}, both of Sasaki-Finsler type. On the other hand, the Sasaki-Finsler metric *G* on *TM*° = *TM* \ {0} induces a Riemannian metric *G*
_{ind} on *™*°. We prove that IF* ^{n}* is totally geodesic immersed in IF

^{n+p}if and only if

*G*=

*G*

^{*}=

*G*

_{ind}on

*™*°.

References [1] M. T. K. Abbassi, Note on the classification theorems of g-natural metricson the tangent bundle of a Riemannian manifold ( M, g ), Comment. Math. Univ. Carolin., 45 (2004), no. 4, 591-596. [2] M. T. K. Abbassi, M. Sarih, On some hereditary properties of Riemanniang-natural metrics on tangent bundles of Riemannian manifolds , Differential Geom. Appl., 22 (2005), no. 1, 19-47. [3] M. T. K. Abbassi, M. Sarih, On natural metrics on tangent bundles of Riemannian manifolds , Arch. Math., 41 (2005), 71-92. [4] M. T. K. Abbassi, M. Sarih

References [1] M.T.K. Abbassi, M. Sarih: On Natural Metrics on Tangent Bundles of Riemannian Manifolds. Archivum Mathematicum 41 (2005) 71–92. [2] N. Cengiz, A.A. Salimov: Diagonal lift in the tensor bundle and its applications. Appl. Math. Comput. 142 (2–3) (2003) 309–319. [3] J. Cheeger, D. Gromoll: On the structure of complete manifolds of nonnegative curvature. Ann. of Math. 96 (2) (1972) 413–443. [4] M. Djaa, N.E.H. Djaa, R. Nasri: Natural Metrics on T 2 M and Harmonicity. International Electronic Journal of Geometry 6 (1) (2013) 100–111. [5] M

References [1] K. M. T. Abbassi and G. Calvaruso, g-Natural contact metrics on unit tangent sphere bundles, Monatsh. Math ., 151 (2006), 89–109. [2] K. M. T. Abbassi and M. Sarih, On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds, Diff. Geom. Appl ., 22 (2005), 19–47. [3] K. M. T. Abbassi and M. Sarih, On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math . (Brno), 41 (2005), 71–92. [4] K. M. T. Abbassi and A. Yampolsky, Transverse totally geodesic submanifolds of tangent bundle

## Abstract

By regarding the complex indicatrix as an embedded CR-hypersurface of the holomorphic tangent bundle in a fixed point, we analyze some aspects of the relations between its CR structure and the considered contact structure. Moreover, using the classification of the almost contact metric structures associated with a strongly pseudo-convex CR-structure, of D. Chinea and C. Gonzales, we determine the classes corresponding to the natural contact structure of the complex indicatrix and the new structures obtained under a gauge transformation.

## Abstract

We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.

## Abstract

In the original definition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle *T M* ⊕ *T* M* that is equal to its ortho-complement with respect to the so-called neutral metric on the big tangent bundle. In this paper, instead of the big tangent bundle we consider the vector bundle *E* ⊕ *E**, where E is a Banach Lie algebroid and *E** its dual. Recall that *E** is not in general a Lie algebroid. We define a bilinear and symmetric form on the vector bundle *E* ⊕ *E** and say that a subbundle of it is a Dirac structure if it is equal with its orthocomplement. Our main result is that any Dirac structure that is closed with respect to a type of Courant bracket, endowed with a natural anchor is a Lie algebroid. In the proof the differential calculus on a Lie algebroid is essentially involved. We work in the category of Banach vector bundles.