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VIII.—Riemann Extensions which have Kähler Metrics.

Published online by Cambridge University Press:  14 February 2012

E. M. Patterson
E. M. Patterson
Affiliation:
United College, University of St Andrews.
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Synopsis

Certain types of 2n-dimensional Riemannian spaces admitting parallel fields of null n-planes are studied. These are known as Riemann extensions of conformal, projective or other classes of spaces of affine connection. The circumstances under which a 2n-dimensional Riemannian space admits two non-intersecting parallel fields of null n-planes are also discussed. Such spaces satisfy a condition similar to Kähler's condition in the theory of complex manifolds, and hence are called Kähler spaces. Necessary and sufficient conditions are found for a Kähler space to be a Riemann extension with respect to one of the parallel fields of null n-planes, and canonical forms are found for the metrics in the cases of Riemann extensions of conformal and projective spaces.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 64 , Issue 2 , 1954 , pp. 113 - 126
Copyright
Copyright © Royal Society of Edinburgh 1954

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References

REFERENCES TO LITERATURE

Bochner, S., 1947. “Curvature in Hermitian metric”, Bull. Amer. Math. Soc., 53, 179–195.Google Scholar
Bochner, S., 1948. “Curvature and Betti numbers (I)”, Ann. Math., 49, 379–390.CrossRefGoogle Scholar
Eisenhart, L. P., 1922. “Condition that a tensor be the curl of a vector”, Bull. Amer. Math. Soc., 28, 425–427.CrossRefGoogle Scholar
Eisenhart, L. P., 1926. Riemannian Geometry. Princeton.Google Scholar
Eisenhart, L. P., 1927. Non-Riemannian Geometry. New York.Google Scholar
Patterson, E. M., 1953. “A characterisation of Kähler manifolds in terms of parallel fields of planes”, J. Lond. Math. Soc., 28, 260–269.CrossRefGoogle Scholar
Patterson, E. M., and Walker, A. G., 1952. “Riemann extensions”, Quart. J. Math., Oxford, ser. 2, 3, 19–28.Google Scholar
Ruse, H. S., 1949. “On parallel fields of planes in a Riemannian space”, Quart. J. Math., Oxford, ser. 1, 20, 218–234.Google Scholar
Walker, A. G., 1949. “On parallel fields of partially null vector spaces”, Quart. J. Math., Oxford, ser. 1, 20, 135–145.Google Scholar
Walker, A. G., 1950. “Canonical form for a Riemannian space with a parallel field of null planes”, Quart. J. Math., Oxford, ser. 2, 1, 69–79.Google Scholar
Walker, A. G., 1954. “Riemann extensions of non-Riemannian spaces”, Convegno di Geometria Differenziale, 1953, Rome, 64–70.Google Scholar
Wong, Y. C., 1954. “Fields of parallel planes in affmely connected spaces”, Quart. J. Math., Oxford, ser. 2, 4, 241–253.Google Scholar