arXiv research

New proof shows holomorphic sectional curvature fully determines curvature tensor.

We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using …

It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves. This curvatur…

On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a metric of constant curvature and, for negative Euler characteristic, exhibited a flow from a given metric to a constant cu…

Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this te…

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We give examples of isospectral manifolds with different local geometry including continuous families of isospectral negatively curved manifolds with boundary as well as …

Analytic plane curves determine unique conformal coordinates.

We study the decomposition of the Riemannian curvature R tensor of an almost quaternion-Hermitian manifold under the action of its structure group Sp(n)Sp(1). Using the minimal connection, we show that most components are determined by the intrinsic torsion ξand its covariant derivative \widetilde\nablaξand determine r…

The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.

Study on pseudo-Einstein 3-manifolds, calculating determinant changes under conformal transformations.

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4π$.

Study discrete analog of zeta-determinant maximization on triangulated surfaces.

The paper connects curvature data to polynomial coefficients in gluing formulas.

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relations…

Translation surfaces in Heisenberg group classified by Gauss map determinant.

Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description …

Researchers found all special metrics in 4D for certain curvature functionals.

This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/

We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the classification of all homogeneous metrics with positive curvature (apart from one s…

Minimal surfaces in harmonic conformally flat space are studied.

Study finds all 4D Lie groups with harmonic curvature.

We determine non-Hopf hypersurfaces with constant mean curvature in the complex projective plane which attain equality in a basic inequality between the maximum Ricci curvature and the squared mean curvature.

The Quillen-Bismut-Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut-Freed connection on…

Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.

We explicitly determine tori that have a parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane

Study constant mean curvature surfaces with integrable boundary conditions.

We determine the curvature equations of natural metrics on tangent bundles and radius r tangent sphere bundles S_rM of a Riemannian manifold M. A family of positive scalar curvature metrics on S_rM is found, for any M with bounded sectional curvature and any chosen constant r.

New method uses relative capacities of geodesic balls to determine scalar curvature.

Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered.

In this short note, based on the work of Wang-Zhu, we determine the greatest lower bounds on Ricci curvature for all toric Fano manifolds.

In this paper we form relations for the determination of the elements of the Eötvös matrix of the Earth's normal gravity field. In addition a relation between the Gauss curvature of the normal equipotential surface and the Gauss curvature of the actual equipotential surface both passing through the point P is presented…

We solve the Bonnet problem for surfaces in the homogeneous 3-manifolds with a 4-dimensional isometry group. More specifically, we show that a simply connected real analytic surface in H^2xR or S^2xR is uniquely determined pointwise by its metric and its principal curvatures if and only if it is not a minimal or a prop…

The main purpose of this paper is to determine the admissible forms of the sectional curvature operator on a three-dimensional locally homogeneous Lorentzian manifolds.

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.

We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a complete classification of immersed constant mean curvature spheres in three-dimension…

Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.

Study timelike surfaces with parallel mean curvature in Minkowski 4-space.

Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.

Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.…

Minimal normal curvature immersions in the unit ball studied.

Study determines scalar curvature invariants for 3-spheres embedded in 4-manifolds.

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm $|\nabla R|$ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determi…

Formula derived for spectral determinant of sphere with conical singularities.

Natural metric structures on tangent bundles and tangent sphere bundles enclose many important problems, from the topology of the base to the determination of their holonomy. We make here a brief study of the topic. We find the characteristic classes of some of those structures. We solve the question of when two given …

We determine all helix surfaces with parallel mean curvature vector field, which are not minimal or pseudo-umbilical, in spaces of type $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a simply-connected $n$-dimensional manifold with constant sectional curvature $c$.

We determine the isomorphism classes of symmetric symplectic manifolds of dimension at least 4 which are connected, simply-connected and have a curvature tensor which has only one non-vanishing irreducible component -- the Ricci tensor.

We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.