Metric Of Constant Curvature Research Articles

A classification of locally homogeneous affine connections on compact surfaces

We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi–Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane. This refines previous results by Opozda.

Modular curvature for noncommutative two-tori

Modular curvature for noncommutative two-tori

Metrics with conic singularities and spherical polygons

A spherical n-gon is a bordered surface homeomorphic to a closed disk, with n distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these polygons and enumerate them in the case that two angles at the corners are not multiples of �. The problem is equivalent to classification of some second order linear differential equations with regular singularities, with real parameters and unitary monodromy. MSC 2010: 30C20, 34M03.

Bounding hyperbolic and spherical coefficients of Maass forms

On développe une nouvelle méthode pour majorer les coefficients de Fourier hyperboliques et sphériques des formes de Maass définies par rapport à des réseaux uniformes généraux.

ON THE PROJECTIVE ALGEBRA OF SOME (α, β)-METRICS OF ISOTROPIC S-CURVATURE

In this paper, we study projective algebra, p(M, F), of special (α, β)-metrics. The projective algebra of a Finsler space is a finite-dimensional Lie algebra with respect to the usual Lie bracket. We characterize p(M, F) of Matsumoto and square metrics of isotropic S-curvature of dimension n ≥ 3 as a certain Lie sub-algebra of the Killing algebra k(M, α). We also show that F has a maximum projective symmetry if and only if F either is a Riemannian metric of constant sectional curvature or locally Minkowskian.

Kropina metrics and Zermelo navigation on Riemannian manifolds

The present paper studies globally defined Kropina metrics as solutions of the Zermelo’s navigation problem. Moreover, we characterize the Kropina metrics of constant flag curvature showing that up to local isometry, there are only two model spaces of them: the Euclidean space and the odd-dimensional spheres.

Estimation of the conformal factor under bounded Willmore energy

We prove for closed, orientable surfaces in \(\ \mathbb{R }^3\ \) with Willmore energy less that \(\ 8 \pi - \delta \ \) and whose conformal structures are compactly contained in moduli space that after applying appropriate Mobius transformations the conformal factors between the induced metrics and conformal metrics of constant curvature are uniformly bounded by constants depending only on \(\ \delta > 0,\) the genus of the surfaces and the compact subset of the moduli space. Secondly, for a given sequence of closed, orientable surfaces as above, we prove that the conformal factor remains bounded without applying Mobius transformations if and only if no topology is lost. Similar estimates hold in higher codimension.

Ricci Flow and the Determinant of the Laplacian on Non-Compact Surfaces

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 curvature metric along which the determinant increases. The aim of this paper is to perform a similar analysis for the determinant of the Laplacian on a non-compact surface whose ends are asymptotic to hyperbolic funnels or cusps. In that context, we show that the Ricci flow converges to a metric of constant curvature and that the determinant increases along this flow.

A characterization of Finsler metrics of constant flag curvature

Using the notion of C-projectively flatness, we give a new characterization of Finsler metrics of constant flag curvature.

Closed surfaces with bounds on their Willmore energy

The Willmore energy of a closed surface in Rn is the integral of its squared mean curvature, and is invariant under Mobius transformations of Rn . We show that any torus in R3 with energy at most 8π − δ has a representative under the Mobius action for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on δ > 0. An analogous estimate is also obtained for closed, orientable surfaces of fixed genus p ≥ 1 in R3 or R4, assuming suitable energy bounds which are sharp for n = 3. Moreover, the conformal type is controlled in terms of the energy bounds. Mathematics Subject Classification (2010): 53A05 (primary); 53A30, 53C21, 49Q15 (secondary).

New proof of a Calabi’s theorem

A Calabi’s theorem says that on a compact Riemann surface, an extremal metric is a constant scalar curvature metric. In this paper, we use a new method to prove this theorem. Then we give an interesting corollary.

ON A CLASS OF PROJECTIVELY FLAT FINSLER METRICS OF NEGATIVE CONSTANT FLAG CURVATURE

In this paper, we prove a structure theorem for projectively flat Finsler metrics of negative constant flag curvature. We show that for such a Finsler metric if the orthogonal group acts as isometries, then the Finsler metric is a slight generalization of Chern–Shen's construction Riemann–Finsler geometry, Nankai Tracts in Mathematics, Vol. 6 (World Scientific Publishing, Hackensack, NJ, 2005), x+192 pp.

The gradient flow of the [formula omitted] curvature energy near the round sphere

We investigate the low-energy behavior of the gradient flow of the L2 norm of the Riemannian curvature on four-manifolds. We show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive Yamabe constant and small initial energy.

Explicit formula for the spectral counting function of the Laplace operator on a compact Riemannian surface of genus g > 1

Let the standard Riemannian metric of constant curvature K = −1 be given on a compact Riemannian surface of genus g > 1. Under this condition, for a class of strictly hyperbolic Fuchsian groups, we obtain an explicit expression for the spectral counting function of the Laplace operator in the form of a series over the zeros of the Selberg zeta function.

$g$-natural metrics of constant curvature on unit tangent sphere bundles

We completely classify Riemannian $g$-natural metrics of constant sectional curvature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$. Since the base manifold $M$ turns out to be necessarily two-dimensional, weaker curvature conditions are also investigated for a Riemannian $g$-natural metric on the unit tangent sphere bundle of a Riemannian surface.

Regularity of the geodesic equation in the space of Sasakian metrics

In this paper, we study a geodesic equation in the space of Sasakian metrics H. The equation leads to the Dirichlet problem of a complex Monge–Ampère type equation on the Kähler cone. This equation differs from the standard complex Monge–Ampère equation in a significant way, with gradient terms involved in the (1,1) symmetric tensor of the operator. We establish appropriate regularity estimates for this complex Monge–Ampère type equation. As geometric application, we show that the space of Sasakian metrics H is a metric space, and the constant transversal scalar curvature metric realizes the global minimum of K-energy if the first basic Chern class C1B⩽0. We also prove that the constant transversal scalar curvature metric is unique in each basic Kähler class if the first basic Chern class is either strictly negative or zero.

Sharp Asymptotics for Toeplitz Determinants and Convergence Towards the Gaussian Free Field on Riemann Surfaces

We consider canonical determinantal random point processes with N particles on a compact Riemann surface X defined with respect to the constant curvature metric. We establish strong exponential concentration of measure type properties involving Dirichlet norms of linear statistics. This gives an optimal central limit theorem (CLT), saying that the fluctuations of the corresponding empirical measures converge, in the large N limit, towards the Laplacian of the Gaussian free field on X in the strongest possible sense. The CLT is also shown to be equivalent to a new sharp strong Szego-type theorem for Toeplitz determinants in this context. One of the ingredients in the proofs are new Bergman kernel asymptotics providing exponentially small error terms in a constant curvature setting.

The Liouville equation in an annulus

We give the solutions to the Liouville equation in an annulus A of R 2 that satisfy a certain Neumann condition on each component of ∂ A . As a consequence, we classify all the metrics of constant curvature in A that have constant geodesic curvature on ∂ A .

Matsumoto Metrics of Constant Flag Curvature: A Puny Class of Finsler Metrics with Constant Curvature

The local structure of Finsler metrics of constant flag curvature have been historically mysterious. It is proved that every Matsumoto metric of constant flag curvature on a closed n-dimensional manifold of dimension n ≥ 3 is either Riemannian or locally Minkowskian.

On the Projective Algebra of Randers Metrics of Constant Flag Curvature

The collection of all projective vector fields on a Finsler space $(M, F)$ is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by $p(M,F)$ and is the Lie algebra of the projective group $P(M,F)$. The projective algebra $p(M,F=\alpha+\beta)$ of a Randers space is characterized as a certain Lie subalgebra of the projective algebra $p(M,\alpha)$. Certain subgroups of the projective group $P(M,F)$ and their invariants are studied. The projective algebra of Randers metrics of constant flag curvature is studied and it is proved that the dimension of the projective algebra of Randers metrics constant flag curvature on a compact $n$-manifold either equals $n(n+2)$ or at most is $\frac{n(n+1)}{2}$.