Özdeğer, Abdülkadir On sectional curvatures of a Weyl manifold. (English) Zbl 1125.53030 Proc. Japan Acad., Ser. A 82, No. 8, 123-125 (2006). In [H. Pedersen and K. P. Tod, Adv. Math. 97, No. 1, 74–109 (1993; Zbl 0778.53041)], it is proved that if \(M\) is a compact positive-definite Einstein-Weyl manifold whose scalar curvature is everywhere strictly negative, then \(M\) is conformal to an Einstein manifold. In this paper, the author shows that if \(M\) is a Weyl manifold, of dimension \(>2\), such that at each point the sectional curvature is independent of the plane chosen, then \(M\) is locally conformal to an Einstein manifold. Reviewer: Marisa Fernandez (Bilbao) Cited in 3 Documents MSC: 53C20 Global Riemannian geometry, including pinching 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) Keywords:Weyl manifold; Einstein manifold; Einstein-Weyl space; sectional curvature Citations:Zbl 0778.53041 × Cite Format Result Cite Review PDF Full Text: DOI Euclid References: [1] V. Hlavaty, Theorie d’immersion d’une \(W_m\) dans \(W_n\), Ann. Soc. Polon. Math., 21 (1949), 196-206. · Zbl 0038.34801 [2] A. Norden, Affinely Connected Spaces , Nauka, Moscow, 1976. · Zbl 0925.53007 [3] G. Zlatanov and B. Tsareva, On the geometry of the nets in the \(n\)-dimensional space of Weyl, J. Geom. 38 (1990), no.1-2, 182-197. · Zbl 0716.53023 · doi:10.1007/BF01222903 [4] J. L. Synge and A. Schild, Tensor Calculus , Univ. Toronto Press, Toronto, Ont., 1949. [5] E. Canfes and A. Özdeğer, Some applications of prolonged covariant differentiation in Weyl spaces, J. Geom., 60 (1997), no.1-2, 7-16. · Zbl 0889.53022 · doi:10.1007/BF01252214 [6] A. Özdeğer, Conformal mapping of Einstein-Weyl spaces and the generalized Einstein’s tensor. (Submitted). [7] D. Lovelock and H. Rund, Tensors, differential forms, and variational principles, Dover publ. Inc., New York, 1989. · Zbl 0308.53008 [8] N. J. Hitchin, Complex manifolds and Einstein’s equations, in Twistor geometry and nonlinear systems ( Primorsko , 1980), 73-99, Lecture Notes in Math., 970, Springer, Berlin, 1982. · Zbl 0507.53025 · doi:10.1007/BFb0066025 [9] H. Pedersen and K. P. Tod, Three-dimensional Einstein-Weyl geometry, Adv. Math. 97 (1993), no.1, 74-1089. · Zbl 0778.53041 · doi:10.1006/aima.1993.1002 [10] H. Pedersen and A. Swann, Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature, J. Reine Angew. Math. 441 (1993), 99-113. · Zbl 0776.53027 · doi:10.1515/crll.1993.440.99 [11] M. Katagiri, On compact conformally flat Einstein-Weyl manifolds, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), no.6, 104-105. · Zbl 0951.53027 · doi:10.3792/pjaa.74.104 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.