Abstract
We show that, in each odd dimensionn =m 2, there is a class of Grassmann quotient spaces not included in Wolf’s classic solution of the Grassmann space form problem. We classify all of these new Grassmann space forms up to isometry. As an application, we exhibit a pair of compact Einstein manifolds of dimensionm 2 with holonomy groups which are abstractly isomorphic yet not conjugate in the orthogonal group, thus proving that a theorem of Besse cannot be extended to non-simply-connected Einstein manifolds.
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References
Besse, A.: Einstein manifolds. Berlin, Heidelberg. New York: Springer 1987.
Kobayashi, S. and Nomizu, K.: Foundations of differential geometry, Vol. II. New York, London, Sydney: Interscience 1969.
McInnes, B.: Holonomy groups of compact Riemannian manifolds: a classification in dimensions up to ten. J. Math. Phys. 34, 4273–4286 (1993).
McInnes, B.: Examples of Einstein manifolds with all possible holonomy groups in dimensions less than seven. J. Math. Phys. 34, 4287–4304 (1993).
O’Neill, B.: Semi-riemannian geometry. New York: Academic Press 1983.
de Siebenthal, J.: Sur les groupes de Lie compacts non connexes. Commentarii Mathematici Helvetici 31, 41–89 (1956).
Wolf, J. A.: Spaces of constant curvature. Wilmington: Publish or Perish 1984.
Wolf, J. A.: Discrete groups, symmetric spaces, and global holonomy. Amer. J. Math. 84, 527–542 (1962).
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McInnes, B. On space forms of Grassmann manifolds. Manuscripta Math 93, 205–217 (1997). https://doi.org/10.1007/BF02677467
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DOI: https://doi.org/10.1007/BF02677467