The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable li…
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48 results for “Sub-Riemannian Geometry”
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 …
Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.
problem Unified framework for Riemannian and sub-Riemannian geometries.
method Study of gauge metric measure spaces.
result Unified synthetic Ricci curvature lower bounds for both Riemannian and sub-Riemannian structures.
Sub-Riemannian geometry connects bike paths to mathematical curves.
problem Understanding bike paths and their mathematical properties.
method Relating sub-Riemannian geometry to bicycle motion and curve shapes.
result Geodesics in sub-Riemannian geometry correspond to specific bike paths.
The Prytz connectionsmath.DG
Mathematical analysis of Prytz planimeter using sub-Riemannian geometry.
problem Historical use of Prytz planimeter to approximate areas.
method Sub-Riemannian geometry and connections/horizontal lifts.
result Mathematical description and analysis of Prytz planimeter.
Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalisation of the la…
Study shows not all smooth paths are optimal in certain geometric structures.
problem Existence of non-smooth sub-Riemannian minimizing geodesics.
method Constructed a but not length-minimizer example.
result Found a real-analytic sub-Riemannian structure with non-smooth minimizers.
Generalizes Connes's tangent groupoid for sub-Riemannian geometry.
problem Calculating tangent cones in sub-Riemannian geometry.
method Constructs a completion of using sub-Riemannian metric.
result Calculates all tangent cones in Gromov-Hausdorff distance.
Spirals are not shortest paths in certain sub-Riemannian geometries.
problem Nonminimality of spiral-like curves in sub-Riemannian manifolds.
method Construction of a competing curve to demonstrate non-minimality.
result Spiral-like curves are not length minimizing in sub-Riemannian manifolds.
Harmonic maps studied in sub-Riemannian geometry for Lie groups.
problem Defining and characterizing harmonic maps in sub-Riemannian settings.
method Generalization of Riemannian harmonic maps to sub-Riemannian manifolds and Lie groups.
result Conditions for sub-Riemannian harmonic maps and their classification.
Study on homogeneous geodesics in sub-Riemannian geometry.
problem Characterizing and understanding homogeneous geodesics in sub-Riemannian manifolds.
method Criterion for geodesics to be homogeneous, proof of geodesic orbit spaces, examples of geodesic orbit sub-Riemannian manifolds.
result Existence of at least one homogeneous geodesic under broad conditions.
New examples of sub-Riemannian structures satisfying Minimizing Sard conjecture found.
problem Finding complete sub-Riemannian structures satisfying the Minimizing Sard conjecture.
method Techniques from nonsmooth analysis and geometric measure theory.
result Complete sub-Riemannian structures associated with distributions of co-rank 2 or generic distributions of rank ≥ 2 satisfy the Minimizing Sard conjecture.
Based on the notion of dilatation structure arXiv:math/0608536, we give an intrinsic treatment to sub-riemannian geometry, started in the paper arXiv:0706.3644 . Here we prove that regular sub-riemannian manifolds admit dilatation structures. From the existence of normal frames proved by Bellaiche we deduce the rest of…
The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
Proves a theorem in sub-Riemannian geometry using Carnot groups.
problem Proving a maximum modulus theorem in sub-Riemannian geometry.
method Using nontrivial counterexamples and analysis in Carnot groups.
result The theorem is best possible, with specific gradient restrictions.
We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Rieman…
Study geodesics in sub-Riemannian manifolds, resolving open questions.
problem Understanding geodesics in sub-Riemannian geometry, especially those that lose regularity.
method Constructing examples and using a lifting procedure.
result Existence of non-smooth and branching minimizing geodesics in real-analytic sub-Riemannian manifolds and Carnot groups.
Introduces canonical connections for sub-Riemannian manifolds with constant symbol.
problem Equivalence problem in sub-Riemannian geometry.
method Introduces canonical grading and compatible affine connection.
result Completely computed structures for contact manifolds of constant symbol.
Paper proves inequalities for forms on sub-Riemannian manifolds.
problem Establishing inequalities for differential forms on sub-Riemannian contact manifolds.
method Using structure of Rumin's complex, Sobolev-Gaffney inequality for Heisenberg groups, and geometric properties.
result Gaffney type inequality in Sobolev spaces for differential forms on sub-Riemannian contact manifolds with bounded geometry.
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 on heat content for submanifolds in sub-Riemannian geometry.
problem Understanding heat content for submanifolds in sub-Riemannian geometry.
method Existence of smooth tubular neighborhood, definition of relative heat content, approximation via smooth neighborhoods, asymptotic expansion analysis.
result Approximation of relative heat content fails to recover the exact expansion.
Sharp proof of sub-Riemannian length-minimizing curves being at least
problem Smoothness of sub-Riemannian length-minimizing curves
method Study of a class of sub-Riemannian structures, proving regularity
result Theorem 1.1 in [6] is sharp
Failure of the measure contraction property via quotients in higher-step sub-Riemannian structuresmath.DG
New sub-Riemannian structures fail synthetic curvature bounds.
problem Failure of synthetic curvature bounds in sub-Riemannian geometry.
method New stability results for local MCP under quotients, applied to specific sub-Riemannian structures.
result Ideal sub-Riemannian structures can fail the MCP, generically for high dimensions and rank > 3.
Notes on sub-Riemannian geometry equivalence problem.
problem Isometric sub-Riemannian manifolds equivalence.
method Introduction to connections, frame bundles, and sub-Riemannian geometry; description of canonical grading and connection.
result Minimal set of isometries for Engel (2,3,4)-manifolds.
New optimality conditions for sub-Riemannian geodesics derived.
problem Optimality conditions for sub-Riemannian geodesics.
method Geometric translation and ODE derivation.
result New second-order necessary optimality conditions.
We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained f…
Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.
problem Understanding magnetic flows on 3D contact sub-Riemannian manifolds.
method Introducing horizontal magnetic flows via closed Rumin differential two-forms and analyzing the lifted sub-Riemannian structure.
result Horizontal magnetic flows can be interpreted as geodesic flows on a suitably lifted structure, which is of Engel type when the magnetic field is non-vanishing.
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.
We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case. A key role in all results is played by the presence of abnormal geodesi…
We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.
We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of [Ohsawa, 2015], providing simple and elegant characterizations of normal and abnormal sub-Riemannian extremals.
There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as…
The abstract discusses how sub-Riemannian manifolds can have branching geodesics.
problem The existence of branching geodesics in sub-Riemannian geometry.
method Analyzing the rank discontinuity of normal geodesics and constructing specific examples.
result Sub-Riemannian manifolds can contain branching normal minimizing geodesics.
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma calculus and --Bochner's formula.
result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
The paper introduces a method for dimension reduction using sub-Riemannian geometry.
problem Dimension reduction for manifold learning and surface reconstruction.
method Combining local linear approximations of a point cloud to obtain lower dimensional bundles.
result Sub-Riemannian geodesics can successfully be applied to problems like constructing an approximating submanifold and computing distances.
The paper extends the functional geometry of the visual cortex to more complex architectures using contactization and symplectization.
problem Understanding the functional architecture of the visual cortex.
method Contactization and symplectization processes to extend the dimension of the space.
result Extension of the functional geometry of the visual cortex to more complex architectures.
We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isola…
In this letter we exhibit the relation between the isometries of a Riemannian contraction of a sub-Riemannian manifold and those of the sub-Riemannian metric, for to use this relation with two goals: establishing a result about the existence of fixed points of isometries groups; and the other, defining a Multiresolutio…
New model explains visual illusions using sub-Riemannian geometry.
problem Mathematical modeling of Poggendorff-type visual illusions.
method Cortical-inspired sub-Riemannian model with sub-Laplacian evolution.
result Sub-Riemannian kernel enhances visual misperceptions and biases.
New method for sampling diffusion bridges on sub-Riemannian manifolds.
problem Sampling conditioned diffusion processes on sub-Riemannian manifolds is challenging.
method Score matching for machine learning, adapted to non-holonomic frames.
result Demonstrated method works on Heisenberg group and other sub-Riemannian manifolds.
Innovative advances validate a conjecture on maximal hypoellipticity in sub-Riemannian geometry.
problem Characterizing maximal hypoellipticity in sub-Riemannian geometry.
method Generalization of Connes tangent groupoid, pseudodifferential calculus, and invertibility of principal symbol.
result Validation of Helffer and Nourrigat's conjecture on maximal hypoellipticity.
Tangency sets of non-involutive distributions and unrectifiability in Carnot-Carathéodory spacesmath.DG
New insights into integrability and rectifiability in sub-Riemannian geometry.
problem Understanding rectifiability in sub-Riemannian spaces.
method Refined Frobenius Theorem for non-involutive distributions, new metric space class.
result Carnot-Carathéodory spaces are extremal in rectifiability.
Tubes in sub-Riemannian geometry and a Weyl's invariance result for curves in the Heisenberg groupsmath.DG
The paper studies sub-Riemannian geometry and proves a Weyl's invariance result for Heisenberg groups.
problem Optimal regularity and volume asymptotics of submanifolds in sub-Riemannian structures.
method Analyzes tubular neighborhoods and uses Weyl's invariance for Heisenberg groups.
result Volume of small tubes around curves in Heisenberg groups is invariant to embedding.
We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems,…
Study holonomy in pseudo-Hermitian geometry structures.
problem Holonomy classification in pseudo-Hermitian geometry.
method Analyzes sub-Riemannian structures, torsion, and holonomy algebras.
result Holonomy groups of Schouten and adapted connections are related under certain conditions.
Study curvature invariants in sub-Riemannian manifolds.
problem Understand curvature invariants in sub-Riemannian geometry.
method Prove geometrical inequalities for submanifolds with orthogonal distributions.
result Inequalities for submanifolds with orthogonal distributions are derived.
Study large deviations in random walks on Lie groups.
problem Large deviations in sub-Riemannian random walks.
method Prove large deviation principle for random walks on stratified Lie groups.
result Proved a large deviation principle with a rate function adapted to sub-Riemannian geometry.
The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere originating from different constructions. Namely, we describe the sub-Riemannian geometry of arising through its right Lie group action over itself, the one inherited from the natural complex…