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In this note we show that any complete Kähler (immersed) Euclidean hypersurface $M^{2n}\subset {ℝ}^{2n+1}$ must be the product of a surface in ${ℝ}^3$ with an Euclidean factor ${ℂ}^{n-1}\cong {ℝ}^{2n-2}$.

We explore some aspects of the topology of the family of 13-dimensional Bazaikin spaces. Using the computation of their homology rings, Pontryagin classes and linking forms, we show that there is only one Bazaikin space that is homotopy equivalent to a homogeneous space, i.e., the Berger space. Moreover, it is easily shown that there are only finitely many Bazaikin spaces in each homeomorphism type and that there are only finitely many positively curved ones for a given cohomology ring. In fact,...

We show that any real Kähler Euclidean submanifold $f:M^{2n}\to {ℝ}^{2n+p}$ with either non-negative Ricci curvature or non-negative holomorphic sectional curvature has index of relative nullity greater than or equal to $2n-2p$. Moreover, if equality holds everywhere, then the submanifold must be a product of Euclidean hypersurfaces almost everywhere, and the splitting is global provided that $M^{2n}$ is complete. In particular, we conclude that the only real Kähler submanifolds $M^{2n}$ in ${ℝ}^{3n}$ that have either positive Ricci curvature or...