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In this paper, we study the positivity property of the tangent bundle $T_X$ of a Fano threefold X with Picard number $2$ . We determine the bigness of the tangent bundle of the whole $36$ deformation types. Our result shows that $T_X$ is big if and only if $(-K_X)^3\\ge 34$ . As a corollary, we prove that the tangent bundle is not big when X has a standard conic bundle structure with non-empty discriminant. Our main methods are to produce irreducible effective divisors on ${\\mathbb {P}}(T_X)$ constructed from the total dual VMRT associated to a family of rational curves. Additionally, we present some criteria to determine the bigness of $T_X$ .

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \\to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.

Let X be a complex smooth projective variety such that the exterior power of the tangent bundle [[conjunction].sup.r] [T.sub.X] is nef for some 1 [less than or equal to] r < dim X. We prove that, up to a finite etale cover, X is a Fano fiber space over an Abelian variety. This gives a generalization of the structure theorem of varieties with nef tangent bundle by Demailly, Peternell and Schneider [5] and that of varieties with nef [[conjunction].sup.r] [T.sub.X] by the author [20]. Our result also gives an answer to a question raised by Li, Ou and Yang [15] for varieties with strictly nef [[conjunction].sup.r] [T.sub.X] when r < dim X. Key words: Tangent bundle; exterior power; nef.

Let (M, g) be a Riemannian manifold and T²M be its second order tangent bundle. In this paper, we deal with certain characterizations of F–geodesics (which generalize both classical geodesics and magnetic curves) on the second order tangent bundle T²M and the hypersurface T 1 , 1 2 with respect to some natural metrics.

In the paper a Riemannian structure on the tangent bundle is defined by using a statistical structure (g,∇) on the base manifold. Expressions for various curvatures of the structure are derived. Some rigidity results of the structure are proved. The main goal of the paper is to initiate the study of sphere bundles over statistical manifolds. Basic formulas for the geometry are established. Sphere bundles with small radii over compact manifolds are studied.

In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \\to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

Let$ (M, g) $be an$ n $ -dimensional (pseudo-)Riemannian manifold and$ TM $be its tangent bundle$ TM $equipped with the complete lift metric$ ^{C}g $ . First, we define a Ricci quarter-symmetric metric connection$ \\overline{\\nabla } $on the tangent bundle$ TM $equipped with the complete lift metric$ ^{C}g $ . Second, we compute all forms of the curvature tensors of$ \\overline{\\nabla } $and study their properties. We also define the mean connection of$ \\overline{\\nabla } $ . Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle$ TM $to become a Ricci soliton and a gradient Ricci soliton concerning$ \\overline{\\nabla } $are presented. Finally, we search conditions for the tangent bundle$ TM $to be locally conformally flat with respect to$ \\overline{\\nabla } $ .

We discuss in this note the algebra H^0(X, Sym*TX) for a smooth complex projective variety X . We compute it in some simple examples, and give a sharp bound on its Krull dimension. Then we propose a conjectural characterization of non-uniruled projective manifolds with pseudo-effective tangent bundle.

We study first-order Sobolev spaces on reflexive Banach spaces via relaxation, test plans, and divergence. We show the equivalence of the different approaches to the Sobolev spaces and to the related tangent bundles.

Let (X 2k, F, g) be an almost anti-paraHermitian manifold with an almost paracomplex structure F and a Riemannian metric g and let TX be its tangent bundle with the ciconia metric g˜. The purpose of this paper is divided into two folds. The first one is to examine the curvature properties of the tangent bundle TX with the ciconia metric g˜. The second one is to study conformal vector fields and almost Ricci and Yamabe solitons on the tangent bundle TX according to the ciconia metric g˜.