Abstract
A closed orientable four-dimensional manifoldM 4, whose sectional curvatures satisfy −1 ≤ K ≤ 1, is considered. If the product of the curvatures of orthogonal area elements is nonnegative and if at least at one point all the curvatures are different from zero, then the estimate vol(M 4) > (4/9)π2 is obtained for the volume ofM 4. A theorem on the local structure of a manifold with small volume whose curvatures at every point are of the same sign is established.
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Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 3–8, 1991.
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Aminov, Y.A. A lower bound for the volume of a four-dimensional closed manifold. J Math Sci 69, 825–828 (1994). https://doi.org/10.1007/BF01250808
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DOI: https://doi.org/10.1007/BF01250808