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Register a new userHermitian threefolds with vanishing real bisectional curvature
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We examine the class of compact Hermitian manifolds with constant holomorphic sectional curvature. Such manifolds are conjectured to be Kahler (hence a complex space form) when the constant is non-zero and Chern flat (hence a quotient of a complex Lie group) when the constant is zero. The conjecture is known in complex dimension two but open in higher dimensions. In this paper, we establish a partial solution in complex dimension three by proving that any compact Hermitian threefold with zero real bisectional curvature must be Chern flat. Real bisectional curvature is a curvature notion introduced by Xiaokui Yang and the second named author in 2019, generalizing holomorphic sectional curvature. It is equivalent to the latter in the Kahler case and is slightly stronger in general.
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In this article we continue the study of the two curvature notions for Kahler manifolds introduced by the first named author earlier: the so-called cross quadratic bisectional curvature (CQB) and its dual ($^d$CQB). We first show that compact Kahler manifolds with CQB$_1>0$ or $mbox{}^d$CQB$_1>0$ are Fano, while nonnegative CQB$_1$ or $mbox{}^d$CQB$_1$ leads to a Fano manifold as well, provided that the universal cover does not contain a flat de Rham factor. For the latter statement we employ the Kahler-Ricci flow to deform the metric. We conjecture that all Kahler C-spaces will have nonnegative CQB and positive $^d$CQB. By giving irreducible such examples with arbitrarily large second Betti numbers we show that the positivity of these two curvature put no restriction on the Betti number. A strengthened conjecture is that any Kahler C-space will actually have positive CQB unless it is a ${mathbb P}^1$ bundle. Finally we give an example of non-symmetric, irreducible Kahler C-space with $b_2>1$ and positive CQB, as well as compact non-locally symmetric Kahler manifolds with CQB$<0$ and $^d$CQB$<0$.
We prove that a complete noncompact K{a}hler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {bf C}$^{n}$ and we show that the manifold is topologically {bf R}$^{2n}$. In particular, when $M^{n}$ is a K{a}hler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to {bf C}$^{2}$.
Every Riemannian metric or Finsler metric on a manifold induces a spray via its geodesics. In this paper, we discuss several expressions for the X-curvature of a spray. We show that the sprays obtained by a projective deformation using the S-curvature always have vanishing X-curvature. Then we establish the Beltrami Theorem for sprays with X=0
In this paper, we study one of the open problems in Finsler geometry which presented by Matsumoto-Shimada about the existence of P-reducible metric which is not C-reducible. For this aim, we study a class of Finsler metrics called generalized P-reducible metrics that contains the class of P-reducible metrics. We prove that every generalized P-reducible $(alpha, beta)$-metric with vanishing S-curvature reduces to a Berwald metric or C-reducible metric. It results that there is not any concrete P-reducible $(alpha,beta)$-metric with vanishing S-curvature.
We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant *-scalar curvature which satisfies the Kaehler condition at the point in question.
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