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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 IFn = (M, F) be a Finsler submanifold of a Finsler manifold IFn+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 IFn is totally geodesic immersed in IFn+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 MT* 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 EE*, 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 EE* 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.