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Monar, Maria Dolores; Vanhecke, Lieven

Locally symmetric quaternionic Kähler manifolds. (English) Zbl 0797.53029

Differ. Geom. Appl. 4, No. 2, 127-149 (1994).
Summary: One derives a criterion for a quaternionic Kähler manifold to be locally symmetric by using the covariant derivative of the Riemannian curvature tensor. Then one shows the usefulness of this fundamental result by characterizing this class of spaces by means of the intrinsic and intrinsic geometry of the geodesic spheres and the properties of local reflections (geodesic symmetries) and rotations. The results are similar to the ones obtained in [M. Djorić and the second author, Math. J. Okoyama Univ. 32, 187-206 (1990; Zbl 0735.53049) and K. Sekigawa and the second author, Q. J. Math., Oxf. II. Ser. 37, 95-103 (1986; Zbl 0589.53068)] in the framework of Kähler geometry.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C35 Differential geometry of symmetric spaces
53C40 Global submanifolds

Keywords:

quaternionic Kähler manifolds; locally symmetric spaces; quaternionic space forms; quaternionic transformations; geodesic spheres; local reflections; local rotations

Citations:

Zbl 0735.53049; Zbl 0589.53068
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Full Text: DOI

References:

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