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Sinhwi Kim

Publications and source records attributed to Sinhwi Kim.

7 recordsLinked to original sources
arXiv2026

Scalar-flat K\"ahler surfaces whose Weyl tensor annihilates the Ricci form

We conjecture that any scalar-flat K\"ahler surface in which the Weyl tensor acting on 2-forms annihilates the Ricci form must be either Ricci-flat or locally isometric to a Riemannian product of two real surfaces with mutually opposite nonzero constant Gaussian curvatures. This amounts to the nonexistence of proper weakly Einstein anti-self-dual K\"ahler surfaces. We prove the above conjecture in three special cases: when the manifold is compact, when one of the Ricci eigendistributions is integrable, and when the norms of the Ricci and Weyl tensors are functionally dependent

math.DG↗
arXiv2024

On weakly Einstein K\"ahler surfaces

Riemannian four-manifolds in which the triple contraction of the curvature tensor against itself yields a functional multiple of the metric are called weakly Einstein. We focus on weakly Einstein K\"ahler surfaces. We provide several conditions characterizing those K\"ahler surfaces which are weakly Einstein, classify weakly Einstein K\"ahler surfaces having some specific additional properties, and construct new examples.

math.DG↗
arXiv2024

On weakly Einstein Lie groups

A Riemannian manifold is called \emph{weakly Einstein} if the tensor $R_{iabc}R_{j}^{~~abc}$ is a scalar multiple of the metric tensor $g_{ij}$. We consider weakly Einstein Lie groups with a left-invariant metric which are weakly Einstein. We prove that there exist no weakly Einstein non-abelian $2$-step nilpotent Lie groups and no weakly Einstein non-abelian nilpotent Lie groups whose dimension is at most $5$. We also prove that an almost abelian Lie group is weakly Einstein if and only if at the Lie algebra level it is defined by a normal operator whose square is a multiple of the identity.

math.DG↗
arXiv2022

Two theorems on the intersections of horospheres in asymptotically harmonic spaces

We use Busemann functions to construct volume preserving mappings in an asymptotically harmonic manifold. If the asymptotically harmonic manifold satisfies the visibility condition, we construct mappings which preserve distances in some directions. We also prove that some integrals on the intersection of horospheres are independent of the differences between the values of the corresponding Busemann functions and we establish an upper bound of the volume of the intersection of two horospheres which is independent of the difference between values of corresponding Busemann functions.

math.DG↗
arXiv2021

Einstein hypersurfaces of Damek-Ricci spaces

Einstein hypersurfaces are ``very rare" in rank-one symmetric spaces. Damek-Ricci spaces may be viewed as the closest and the most natural generalisations of noncompact rank-one symmetric spaces. We prove that no Damek-Ricci space admits an Einstein hypersurface.

math.DG↗
arXiv2019

Totally geodesic submanifolds of Damek-Ricci spaces and Einstein hypersurfaces of the Cayley projective plane

We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic manifold is by itself harmonic. We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a particular radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces. We also show that if a $2$-stein space admits a $2$-stein hypersurface, then both are of constant curvature, under some additional conditions.

math.DG↗