Summary
Locally product complex spaces are introduced with particular reference to the Calculus of Variations on complex manifolds. The goedesics of such spaces possess unusual features associated with the fact that many of the connection coefficients are tensorial in character. A restricted partial covariant derivative is introduced together with a curvature tensor. The latter is found to be invariant under a large class of gauge-like transformations. This invariance property leads naturally to the introduction of almost totally decomposable complex spaces. Necessary and sufficient conditions for a locally product complex Riemannian space to be almost totally decomposable are discussed. The counterpart of the Einstein tensor is also exhibited.
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Literature
S. Bochner,Vector Fields and Ricci Curvature. Bull. Amer. Math. Soc.52, 776–797 (1946).
S. Bochner, andW. T. Martin,Several Complex Variables. Princeton University Press, Princeton (1948).
D. Lovelock,Complex Spaces with Locally Product Metrics: Special Spaces.
H. Rund,Calculus of Variations on Camplex Manifolds. Third Colloquium on the Calculus of Variations, University of South Africa, 10–85 (1965).
H. Rund,The Differential Geometry of Finsler Spaces. Springer-Verlag, Berlin-Göttingen-Heidelberg (1959).
S. Tachibana,Some Theorems on Locally Product Riemannian Spaces. Tôhuku Math. J.12, 281–292 (1960).
K. Yano,Differential Geometry on Complex and Almost Complex Spaces. Pergamon Press, Oxford (1965).
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On leave of absence from the Department of Mathematics, The University. Bristol.
Entrata in Redazione il 30 ottobre 1968.
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Lovelock, D. Complex spaces with locally product metrics: general theory. Annali di Matematica 83, 53–72 (1969). https://doi.org/10.1007/BF02411160
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DOI: https://doi.org/10.1007/BF02411160