Sectional Curvature in 4-Dimensional Manifolds

Graham Hall

Abstract: We consider the sectional curvature function on a 4-dimensional manifold admitting a metric $g$ of neutral signature, $(+,+,-,-)$ together with a review of the situation for the other two signatures. The main results of the paper are: first, that if the sectional curvature function is not a constant function at any $m\in M$ (actually a slightly weaker assumption is made), the conformal class of $g$ is always uniquely determined and in almost all cases $g$ is uniquely determined on $M$, second, a study of the special cases when this latter uniqueness does not hold, third, the construction of the possible metrics in this latter case, fourth, some remarks on sectional curvature preserving vector fields and finally the complete solution when $(M,g)$ is Ricci flat.

Keywords: sectional curvature, 4-dimensions, neutral signature

Classification (MSC2000): 53A07; 53A35

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