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Register a new userThe Holomorphic Sectional Curvature of General Natural KAhler Structures on Cotangent Bundles
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S. L. Druta
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We study the conditions under which a Kahlerian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ has constant holomorphic sectional curvature. We obtain that a certain parameter involved in the condition for $(T^*M,G,J)$ to be a Kahlerian manifold, is expressed as a rational function of the other two, their derivatives, the constant sectional curvature of the base manifold $(M,g)$, and the constant holomorphic sectional curvature of the general natural Kahlerian structure $(G,J)$.
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We study the conditions under which an almost Hermitian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ is K ahlerian. First, we obtain the algebraic conditions under which the manifold $(T^*M,G,J)$ is almost Hermitian. Next we get the integrability conditions for the almost complex structure $J$, then the conditions under which the associated 2-form is closed. The manifold $(T^*M,G,J)$ is K ahlerian iff it is almost Kahlerian and the almost complex structure $J$ is integrable. It follows that the family of Kahlerian structures of above type on $T^*M$ depends on three essential parameters (one is a certain proportionality factor, the other two are parameters involved in the definition of $J$).
We study the conditions under which the cotangent bundle $T^*M$ of a Riemaannian manifold $(M,g)$, endowed with a Kahlerian structure $(G,J)$ of general natural lift type (see cite{Druta1}), is Einstein. We first obtain a general natural Kahler-Einstein structure on the cotangent bundle $T^*M$. In this case, a certain parameter, $lambda$ involved in the condition for $(T^*M,G,J)$ to be a Kahlerian manifold, is expressed as a rational function of the other two, the value of the constant sectional curvature, $c$, of the base manifold $(M,g)$ and the constant $rho$ involved in the condition for the structure of being Einstein. This expression of $lambda$ is just that involved in the condition for the Kahlerian manifold to have constant holomorphic sectional curvature (see cite{Druta2}). In the second case, we obtain a general natural Kahler-Einstein structure only on $T_0M$, the bundle of nonzero cotangent vectors to $M$. For this structure, $lambda$ is expressed as another function of the other two parameters, their derivatives, $c$ and $rho$.
We study some properties of the tangent bundles with metrics of general natural lifted type. We consider a Riemannian manifold $(M,g)$ and we find the conditions under which the Riemannian manifold $(TM,G)$, where $TM$ is the tangent bundle of $M$ and $G$ is the general natural lifted metric of $g$, has constant sectional curvature.
We show that a closed almost Kahler 4-manifold of globally constant holomorphic sectional curvature $kgeq 0$ with respect to the canonical Hermitian connection is automatically Kahler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern-Weil theory implies useful integral formulas, which are then combined with results from Seiberg--Witten theory.
We prove that for a bounded domain in $mathbb C^n$ with the Bergman metric of constant holomorphic sectional curvature being biholomorphic to a ball is equivalent to the hyperconvexity or the exhaustiveness of the Bergman-Calabi diastasis. By finding its connection with the Bergman representative coordinate, we give explicit formulas of the Bergman-Calabi diastasis and show that it has bounded gradient. In particular, we prove that any bounded domain whose Bergman metric has constant holomorphic sectional curvature is Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation and characterize pseudoconvex domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set.
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