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Register a new userOn sprays with vanishing X-curvature
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Authors
Zhongmin Shen
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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
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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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