Positive-Ricci-curvature conjecture for parabolic geometries

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Let GG be the split real form of a semisimple Lie group with parabolic subgroup PP, and let P→E→MP\to E\to M be a G/PG/P-geometry. Suppose that the Lie algebra g\mathfrak{g} has its parabolic filtration with p=g0\mathfrak{p}=\mathfrak{g}^0, inducing a subbundle E×PV1⊂TME\times_P V_1\subset TM, where V1V_1 is the first filtered piece. Choose a principal maximal compact-subgroup reduction Ec→ME^c\to M, and let it induce the canonical Riemannian metric on MM. Positive-Ricci completeness conjecture. If this Riemannian metric has positive Ricci curvature along E×PV1E\times_P V_1, then the G/PG/P-geometry is complete. The claim proposes a geometric completeness criterion based on Ricci curvature in the distinguished filtered directions; the supplied text gives no resolution.

References

Primary source

Benjamin McKay, “Morphisms of Cartan connections”, arXiv:0802.1473 (2010).

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