Abstract
In this short note we introduce a new center of mass argument on the space of complex structures on any tangent space of a Riemannian manifold. We use this to show there is a universal positive constant, depending only on the dimension of the manifold, according to which an almost hermitian manifold is Kähler if and only if the holonomy action, induced on the space of complex structures at a point, is preserved up to error given by this constant.
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References
Chavel, I.: Riemannian Geometry, Volume 98 of Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2006). (A modern introduction)
Cirici, J., Wilson, S.O.: Dolbeault cohomology for almost complex manifolds. Preprint arxiv:1809.1416 (2018)
Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)
Frolicher, A.: Relations between the cohomology groups of Dolbeault and topological invariants. Proc. Natl. Acad. Sci. USA 41, 641–644 (1955)
Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30(5), 509–541 (1977)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Interscience Tracts in Pure and Applied Mathematics, vol. II, No. 15. Wiley, New York (1969)
Sullivan, D.: On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Volume 97 of Annals of Mathematics, pp. 465–496. Princeton University Press, Princeton, NJ (1981)
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Wilson, S.O., Zeinalian, M. Center of mass and Kähler structures. J. Geom. 110, 33 (2019). https://doi.org/10.1007/s00022-019-0489-8
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DOI: https://doi.org/10.1007/s00022-019-0489-8