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JeongHyeong Park

Publications and source records attributed to JeongHyeong Park.

At least 37 records · Page 2Linked to original sources
arXiv2021

Einstein hypersurfaces in irreducible symmetric spaces

We show that if $M$ is an Einstein hypersurface in an irreducible Riemannian symmetric space $\overline{M}$ of rank greater than $1$ (the classification in the rank-one case was previously known), then either $\overline{M}$ is of noncompact type and $M$ is a codimension one Einstein solvmanifold, or $\overline{M}=\mathrm{SU}(3)/\mathrm{SO}(3)$ (respectively, $\overline{M}=\mathrm{SL}(3)/\mathrm{SO}(3)$) and $M$ is foliated by totally geodesic spheres (respectively, hyperbolic planes) of $\overline{M}$, with the space of leaves parametrised by a special Legendrian surface in $S^5$ (respectively, by a proper affine sphere in $\mathbb{R}^3$).

math.DG↗
arXiv2021

Centrally Harmonic spaces

We construct examples of centrally harmonic spaces by generalizing work of Copson and Ruse. We show that these examples are generically not centrally harmonic at other points. We use this construction to exhibit manifolds which are not conformally flat but such that their density function agrees with Euclidean space.

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↗
arXiv2020

Harmonic spaces

We use the density function of a harmonic space to obtain estimates for the eigenvalues of the Jacobi operator; when these estimates are sharp, then the harmonic space is a symmetric Osserman space.

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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↗
arXiv2018

Almost h-conformal slant submersions from almost quaternionic Hermitian manifolds

We introduce the notions of h-conformal slant submersions and almost h-conformal slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, h-slant submersions, almost h-slant submersion and conformal slant submersions. We investigate several properties of these including the integrability of distributions, the geometry of foliations and the conditions for such maps to be totally geodesic. Further we give some examples of such maps.

math.DG↗
arXiv2017

Hessian of Busemann functions and rank of Hadamard manifolds

In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to eigenvalue zero is investigated with respect to rank of geodesics. On a harmonic Hadamard manifold which is of purely exponential volume growth, or of hypergeometric type it is shown that every Busemann function admits positive definite Hessian. A criterion for $(M,g)$ fulfilling visibility axiom is presented in terms of positive definiteness of the Hessian of Busemann functions.

math.DG↗
arXiv2016

Moduli spaces of oriented Type A manifolds of dimension at least 3

We examine the moduli space of oriented locally homogeneous manifolds of Type A which have non-degenerate symmetric Ricci tensor both in the setting of manifolds with torsion and also in the torsion free setting where the dimension is at least 3. These exhibit phenomena that is very different than in the case of surfaces. In dimension 3, we determine all the possible symmetry groups in the torsion free setting.

math.DG↗
arXiv2015

Euler-Lagrange formulas for pseudo-Kaehler manifolds

Let $c$ be a characteristic form of degree $k$ which is defined on a Kaehler manifold of real dimension $m>2k$. Taking the inner product with the Kaehler form $Ω^k$ gives a scalar invariant which can be considered as a generalized Lovelock functional. The associated Euler-Lagrange equations are a generalized Einstein-Gauss-Bonnet gravity theory; this theory restricts to the canonical formalism if $c=c_2$ is the second Chern form. We extend previous work studying these equations from the Kaehler to the pseudo-Kaehler setting.

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arXiv2012

Spectral geometry of $eta$-Einstein Sasakian manifolds

We extend a result of Patodi for closed Riemannian manifolds to the context of closed contact manifolds by showing the condition that a manifold is an $η$-Einstein Sasakian manifold is spectrally determined. We also prove that the condition that a Sasakian space form has constant $ϕ$-sectional curvature $c$ is spectrally determined.

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arXiv2011

Universal curvature identities II

We show that any universal curvature identity which holds in the Riemannian setting extends naturally to the pseudo-Riemannian setting. Thus the Euh-Park-Sekigawa identity also holds for pseudo-Riemannian manifolds. We study the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that as in the Riemannian setting, they are given solely in terms of curvature (and not in terms of covariant derivatives of curvature) even in the pseudo-Riemannian setting.

math.DG↗
arXiv2011

Hermitian structures on the product of Sasakian manifolds

We investigate the curvature properties of a two-parameter family of Hermitian structures on the product of two Sasakian manifolds, as well as intermediate relations. We give a necessary and sufficient condition for a Hermitian structure belonging to the family to be Einstein and provide concrete examples.

math.DG↗