We obtain some results on symmetries of sub-Riemannian surfaces. In case of contact sub-Riemannian surface we base on invariants found by Hughen \cite{Hughen}. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariant…
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48 results for “Riemannian surfaces”
Study sub-Riemannian surfaces in contact manifolds, proving a Gauss-Bonnet theorem.
problem Proving a Gauss-Bonnet theorem for sub-Riemannian surfaces in contact manifolds.
method Using a family of taming Riemannian metrics, the theorem is derived in the limit.
result Recover topological information of surfaces from geometry around characteristic set.
Study determines a minimal surface in a Riemannian manifold from boundary data.
problem Determining a minimal surface in a Riemannian manifold from boundary data.
method Analyzes the Dirichlet-to-Neumann map for the minimal surface equation.
result Knowledge of the Dirichlet-to-Neumann map determines the Riemannian manifold up to isometry.
The study examines minimal surfaces in Riemannian products of surfaces.
problem Exploring geometric and topological restrictions on minimal surfaces in Riemannian products of surfaces.
method Analyzes totally geodesic surfaces and minimal 2-spheres, 2-tori, and 2-spheres in Riemannian products of surfaces with constant curvature.
result Generically, a totally geodesic surface in a Riemannian product is either a slice or a product of geodesics. Minimal 2-spheres and 2-tori have specific properties under certain curvature conditions.
Study stochastic processes on surfaces in contact sub-Riemannian manifolds using Riemannian approximations.
problem Analyzing stochastic processes on surfaces in contact sub-Riemannian manifolds.
method Employing Riemannian approximations, a second order partial differential operator is derived on the surface. The stochastic process moves along the characteristic foliation induced by the contact distribution.
result Elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from separatrices.
Unified study of surfaces using Clifford algebras.
problem Classifying immersed surfaces in various manifolds.
method Using Clifford algebras to construct formalism for immersed bilegendrian surfaces.
result Full classifications of immersed bilegendrian surfaces in the unit tangent bundle of the 3-sphere.
Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solution…
New inequality links surface orthospectrum to boundary length.
problem Establishing a relationship between orthospectrum and boundary length for compact Riemannian surfaces.
method Analyzing compact Riemannian surfaces with a single closed geodesic, establishing a uniform lower bound on boundary length in terms of orthospectrum.
result A uniform lower bound on boundary length in terms of orthospectrum, akin to Basmajian's identity.
In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian minimal surfaces can be defined as the critical points of the '`horizontal'' ar…
Improved bounds on geodesic lengths in Riemannian surfaces.
problem Finding precise lengths of geodesics in Riemannian surfaces.
method Proved curvature-free linear length bounds on geodesics.
result Length of -shortest geodesic is at most .
We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in to other Lorentzian space forms. We also characterize immersions of Riemannia…
An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in ge…
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove t…
Catenaries defined on any Riemannian surface using intrinsic distance.
problem Defining catenaries on Riemannian surfaces.
method Defining catenaries as critical points of a potential functional, calculating potential with intrinsic distance, and characterizing using curvature.
result Characterization of catenaries on various Riemannian surfaces.
We study the classification of area-stationary and stable regular surfaces in the space of the rigid motions of the Minkowski plane E(1,1), equipped with its sub-Riemannian structure. We construct examples of area-stationary surfaces that are not foliated by sub-Riemannian geodesics. We also prove that there exis…
Study on curvatures of surfaces in specific Lie groups.
problem Analyzing curvatures of surfaces in 3D contact sub-Riemannian Lie groups.
method Riemannian approximation scheme to derive formulas for curvatures.
result Classification of surfaces with constant horizontal curvatures.
Study proves finiteness for distance functions on curved surfaces with controlled curvature.
problem Understanding distance functions on curved surfaces with Hölder continuous curvature.
method Proves a finiteness principle using Whitney extension theory for geodesics and points on Riemannian surfaces with Hölder continuous curvature.
result Establishes a finiteness principle for isometric embedding of metric spaces into Riemannian surfaces with controlled curvature.
Sharp inequalities and symmetries on Riemannian surfaces quantified.
problem Understanding symmetries and asymmetries in Riemannian surfaces.
method Introducing scattering energy to measure asymmetry and proving isoperimetric inequalities.
result Sharp quantitative isoperimetric inequalities and domains with vanishing scattering energy characterized.
The paper classifies helix curves on a pseudo-Riemannian surface.
problem Classifying helix curves on pseudo-Riemannian surfaces.
method Analyzing geodesic flow vector fields and pseudo-Riemannian metrics.
result All helix curves are circular helixes with constant curvature and torsion.
Paper proves every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.
problem Characterizing homogeneous Landsberg surfaces.
method Proved isotropic flag curvature and used it to prove rigidity.
result Every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.
The paper proves a diastolic inequality linking surface area and loop length.
problem Finding short geodesics on Riemannian surfaces.
method Proving a universal inequality between diastole and area of closed surfaces.
result Every Riemannian surface can be decomposed into two domains with bounded boundary length.
Sharp upper bound found for stable minimal surfaces.
problem Bounding the diameter of stable minimal surfaces.
method Analyzing three-dimensional Riemannian manifolds with specific curvature conditions.
result Sharp upper bound for the diameter of stable minimal surfaces.
Develops Riemannian geometry for noncommutative super surfaces.
problem No specific problem stated; focuses on mathematical development.
method Introduces metric and connections on noncommutative super surfaces, showing compatibility and zero torsion under certain conditions.
result Noncommutative super surfaces have a well-defined Riemannian geometry with properties analogous to classical Riemannian geometry.
The study generalizes Minkowski inequalities for curves on surfaces.
problem Generalizing Minkowski inequalities for curves on Riemannian surfaces.
method Introducing a generalized Minkowski average using parametrized curves and proving the inequality for constant-speed curves.
result A family of constant-speed curves on a Riemannian surface satisfies the Brunn-Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature is determined by a function κ satisfying a specific inequality.
We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general relativity.
Extends Newton's minimal resistance problem to Riemannian surfaces.
problem Minimal resistance on Riemannian surfaces.
method Derive resistance functional, analyze constrained minimization.
result Smooth extremals are loxodromes, global minimizers characterized.
For constant mean curvature surfaces of class immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classi…
The study examines Riemannian surfaces with simple singularities.
problem Geometry of Riemannian surfaces with discrete singular points.
method Local and global description using divisors, Gauss-Bonnet formula, classifications theorem.
result Classification of flat metrics with simple singularities on compact surfaces.
Generalized Ricci surfacesmath.DG
The study explores surfaces with curvature satisfying a specific relation, leading to isometric immersions and topological obstructions.
problem Curvature relations on smooth Riemannian surfaces and their geometric implications.
method Properties of log-harmonic functions and isometric immersions theorems.
result Characterization of surfaces that locally admit minimal isometric immersions into constant curvature manifolds.
Author finds the solutions of the Christoffel problem for open and closed surfaces in Riemannian space. The Christoffel problem is reduced to the problem of construction the continuous G-deformations preserving the sum of principal radii of curvature for every point of surface in Riemannian space. G-deformation transfe…
The study proves surfaces with high genus have a specific inequality.
problem Proving surfaces with high genus satisfy a specific inequality.
method Using volume entropy and systolic ratio inequality.
result Every closed surface of genus at least 18 satisfies Loewner's systolic ratio inequality.
We show that isothermic surfaces and S-Willmore surfaces are also the solutions to the corresponding Blaschke's problem for both spacelike and timelike surfaces in pseudo-Riemannian space forms. For timelike surfaces both Willmore and isothermic, we obtain a description by minimal surfaces similar to the classical resu…
Study proves rigidity of capillary surfaces in curved 3D spaces.
problem Proving rigidity of capillary surfaces in curved 3D spaces.
method Local rigidity result for infinitesimally rigid capillary surfaces in Riemannian 3-manifolds with mean convex boundary.
result Bounds on genus, boundary components, and area of compact capillary minimal surfaces with low index.
Two definitions quantify regularity of Riemannian surfaces.
problem Quantify the regularity of Riemannian surfaces.
method Intrinsic and extrinsic definitions using Hölder norms and smooth local representations.
result Intrinsic and extrinsic definitions are equivalent up to a constant.
Proves Steiner and tube formulae for 3D contact sub-Riemannian surfaces.
problem Calculating surface properties in complex geometric structures.
method Develops a local Steiner formula for regular surfaces in 3D contact sub-Riemannian manifolds.
result Establishes a formula for surface expansion in arbitrary regions of contact sub-Riemannian manifolds.
Study on surfaces of genus g≥1 in 3D contact sub-Riemannian manifolds, proving finiteness or infiniteness of induced distance.
problem Determining the finiteness of the induced distance on surfaces of genus g≥1 in 3D contact sub-Riemannian manifolds.
method Analyzing the structural stability of the finiteness/not-finiteness of the induced distance on closed surfaces of genus g≥1.
result Closed surfaces of genus g≥1 can be embedded in such a way that the induced distance is either always finite or always infinite.
Selects points from Jordan domains on Riemannian surfaces.
problem Selecting points from Jordan domains in Riemannian surfaces.
method Fiber bundle theory and conformal mappings.
result Space of Jordan domains retracts onto round disks.
Authors calculate limits of curvatures on surfaces in sub-Riemannian manifolds.
problem Calculating limits of Gaussian and normal curvatures on surfaces in sub-Riemannian manifolds.
method Utilized Riemannian approximations scheme in Heisenberg group to calculate limits of curvatures.
result Obtained Gauss-Bonnet theorem as a limit of theorems in approximations schemes.
Study extends Yang-Mills energy gap to Kähler surfaces.
problem Extend Yang-Mills energy gap to Kähler surfaces.
method Extend -energy gap to compact Kähler surfaces with generic Kähler metrics.
result All ASD connections on principal bundles over Kähler surfaces are irreducible.
Arnlind, Hoppe and Huisken showed how to express the Gauss and mean curvature of a surface embedded in a Riemannian manifold in terms of Poisson brackets of the embedding coordinates. We generalize these expressions to the pseudo-Riemannian setting and derive explicit formulas for the case of surfaces embedded in $\R^m…
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact…
Unified study of harmonic maps between pseudo-Riemannian surfaces.
problem Classifying harmonic maps between pseudo-Riemannian surfaces.
method Unified formalism and Bäcklund transformation.
result Unified solutions to harmonic map equations and corresponding maps.
On the divergence representation of the Gauss curvature of Riemannian surfaces and its applicationsmath.DG
The paper explores how the Gauss curvature of Riemannian surfaces can be represented as the divergence of a vector field.
problem Representing the Gauss curvature of Riemannian surfaces as the divergence of a vector field.
method Investigates the existence of a metric linear connection of zero curvature and its role in differential geometry.
result Provides conditions under which a Riemannian surface can be considered a generalized Berwald surface.
Study Liouville equation on Riemannian surfaces, linking volume growth to classification results.
problem Classifying solutions and manifolds of the Liouville equation on Riemannian surfaces.
method Analyzing the Liouville equation on Riemannian surfaces with non-negative Ricci curvature, considering asymptotic volume growth.
result Established classification results for solutions and manifolds, revealing a connection between volume growth and classification.
Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
problem Analyzing locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
method Examines Riemannian and Finslerian surfaces, providing necessary and sufficient conditions for locally symmetric fourth root metrics in 2D and more complex conditions for higher dimensions.
result Formulates conditions for positive definiteness of locally symmetric polynomial metrics in Finslerian surfaces and provides explicit examples.
In this paper, we consider surfaces in 4--dimensional pseudo--Riemannian space--forms with index 2. First, we obtain some of geometrical properties of such surfaces considering their relative null space. Then, we get classifications of quasi--minimal surfaces with positive relative nullity.
Study Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds.
problem Analyzing the Schrödinger evolution on surfaces embedded in 3D contact sub-Riemannian manifolds.
method Relating self-adjointness of the Schrödinger operator to geometric invariants of the foliation.
result Classification of self-adjoint extensions yielding disjoint dynamics.
Rigidity theorem for discrete metric spaces embedded in Riemannian surfaces.
problem Understanding the rigidity of discrete metric spaces embedded in Riemannian surfaces.
method Proving that certain discrete metric spaces are rigidly embedded in the Euclidean plane or other Riemannian surfaces.
result Riemannian embeddings of certain discrete metric spaces are rigid, meaning they cannot be deformed without changing distances.