Bochner-Kahler metrics
By Robert L. Bryant. 2000
161 citations recorded by the source corpus.
A Kahler metric is said to be Bochner-Kahler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kahler metrics in complex dimension n has real dimension n+1 and a recipe for an explicit formula for any Bochner-Kahler metric will be given. It is shown that any Bochner-Kahler metric in complex dimension n has local (real) cohomogeneity at most n. The Bochner-Kahler metrics that can be `analytically continued' to a complete metric, free of singularities, are identified. In particular, it is shown that the only compact Bochner-Kahler manifolds are the discrete quotients of the known symmetric examples. However, there are compact Bochner-Kahler orbifolds that are not locally symmetric. In fact, every weighted projective space carries a Bochner-Kahler metric. The fundamental technique is to construct a canonical infinitesimal torus action on a Bochner-Kahler metric whose associated momentum mapping has the orbits of its symmetry pseudo-groupoid as fibers.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/4d4fb5665a0ff807ef5150c95f06e600a352d74b
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In Applied Mathematics
- Christina W. Tønnesen-Friedman
- P. Gauduchon
- D. Calderbank
- V. Apostolov
- Charles P. Boyer
- Krzysztof Galicki
- Dmitri Alekseevsky
- C. Medori
In Astronomy and Astrophysics
- V. Apostolov
- Charles P. Boyer
- Dmitri Alekseevsky
- C. Medori
- D. Calderbank
- Christina W. Tønnesen-Friedman
- Christina W. Tonneson-Friedman
- Adriano Tomassini
In Geometry and Topology
- Christina W. Tønnesen-Friedman
- P. Gauduchon
- D. Calderbank
- V. Apostolov
- Charles P. Boyer
- Krzysztof Galicki
- Dmitri Alekseevsky
- C. Medori
Citations in the corpus
- Canonical Sasakian Metrics (2006, background): “ighted projective space CPn w. The space of metrics M(ξw,J¯) associated with the polarized Sasakian manifold (S2n+1,ξ w,J¯) has a representative gw whose transverse Ka¨hler metric gT w is Bochner flat [8] on CPn w, and thus, extremal. Computing in an affine orbifold chart, it can be determined [16] tha”
- Homogeneous para-Kähler Einstein manifolds (2009, background): “Such a structure is called a bi-Lagrangian structure [52] or a Lagrangian 2-web (see [53]).”
- Hamiltonian 2-forms in Kähler geometry, IV weakly Bochner-flat Kähler manifolds (2005, background): “Examples include Bochner-flat Kähler metrics (where the Bochner tensor is zero, see [6] for a classification) — in particular metrics of constant holomorphic sectional curvature (CHSC) — and products of Kähler–Einstein metrics (for which ρ is parallel).”
- Hamiltonian 2-Forms in Kähler Geometry, I General Theory (2002, methodology): “When WK(φ) = 0 this system yields an invariant polynomial—in particular for τ0 6= 0 and W K = 0, such a polynomial was found by Bryant [5] and is the basis for his classification of Bochner-flat Kähler metrics.”
- Hamiltonian 2-forms in Kähler geometry, II Global Classification (2004, methodology): “The former case gives the Fubini–Study metric and its orbifold quotients, while the latter recovers the Bochner-flat examples of [10], which are defined on CPm a0,.”
- Maximally homogeneous para-CR manifolds (2006, unknown): “a decomposition of TM into direct sum of two ω-Lagrangian distributions (see Bryant [3]).”