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- W2951844156 abstract "Let $M^m$ be a compact oriented smooth manifold which admits a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of $M^m$ are zero and its Euler number is nonnegative and even. In particular, $M^m$ has signature zero. Since a non-constant harmonic morphism with one-dimensional fibres gives rise to a circle action we have the following applications: (i) many compact manifolds, for example $CP^{n}$, $K3$ surfaces, $S^{2n}times P_g$ ($ngeq2$) where $P_g$ is the closed surface of genus $ggeq2$ can never be the domain of a non-constant harmonic morphism with one-dimensional fibres whatever metrics we put on them; (ii) let $(M^4,g)$ be a compact orientable four-manifold and $phi:(M^4,g)to(N^3,h)$ a non-constant harmonic morphism. Suppose that one of the following assertions holds: (1) $(M^4,g)$ is half-conformally flat and its scalar curvature is zero, (2) $(M^4,g)$ is Einstein and half-conformally flat, (3) $(M^4,g,J)$ is Hermitian-Einstein. Then, up to homotheties and Riemannian coverings, $phi$ is the canonical projection $T^4to T^3$ between flat tori." @default.
- W2951844156 created "2019-06-27" @default.
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- W2951844156 date "2000-07-24" @default.
- W2951844156 modified "2023-09-27" @default.
- W2951844156 title "Topological restrictions for circle actions and harmonic morphisms" @default.
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