This paper introduces tangent display maps to simplify tangent category theory.
problem The category of smooth manifolds does not admit all pullbacks, complicating tangent category theory.
method Develops tangent display maps as a special class of maps well-behaved with respect to pullbacks.
result Tangent display maps simplify previous work in tangent categories and provide a new way to define open subobjects.
Paper constructs infinitely many tangent functors on diffeological spaces.
problem Tangent spaces in diffeological spaces are not uniquely defined.
method Introduced and constructed infinitely many non-isomorphic tangent functors.
result The choice of tangent functor is not unique outside smooth manifolds.
Study on triviality of tangent and generalized tangent bundles of manifolds.
problem Triviality of tangent and generalized tangent bundles of manifolds.
method Analyzing relations between tangent bundle TM and generalized tangent bundle TM=TM⊕T∗M of manifolds. result The generalized tangent bundle of a parallelizable manifold is trivial, but the converse is not always true.
Sprays on Frechet manifolds connect connections and tangent structures.
problem Characterizing linear symmetric connections on Frechet manifolds.
method Constructing connection maps and linear symmetric connections on tangent and second-order tangent bundles using sprays.
result A bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.
The paper proves Γ-convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.
problem Proving convergence of discrete tangent-point energies to continuous energies and ropelength.
method Using biarc curves and interpolation, the paper proves Γ-convergence of discretized tangent-point energies to the continuous tangent-point energies and ropelength functional. result Discrete almost minimizing biarc curves converge to ropelength minimizers and minimizers of continuous tangent-point energies.
We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections o…
Study of normal and tangent maps to frontals.
problem Understanding geometric and dynamical properties of frontals.
method Geometrical and dynamical analysis of normal and tangent maps.
result Parallels of the tangent map to a frontal curve are right equivalent to the tangent map of a frontal curve under certain conditions.
Generalizes Connes's tangent groupoid for sub-Riemannian geometry.
problem Calculating tangent cones in sub-Riemannian geometry.
method Constructs a completion of MimesMimesR+imes using sub-Riemannian metric. result Calculates all tangent cones in Gromov-Hausdorff distance.
Analytic sets with unique infinite tangent cone are algebraic.
problem Characterizing analytic sets with unique infinite tangent cones.
method Analytic and algebraic set properties, degree of complex algebraic sets.
result Degree of Lipschitz normally embedded sets equals their infinite tangent cone degree.
We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent …
The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.
problem Characterizing and understanding Ricci flows with closed and smooth tangent flows.
method Analyzing ancient and finite-time singularity Ricci flows to prove uniqueness and characterizations.
result The tangent flow is unique and characterizes ancient and finite-time singularity flows.
Uniqueness proven for stable hypersurface tangent cones.
problem Stability and uniqueness of tangent cones for stable hypersurfaces.
method Analysis of isolated singularities and tangent cones of stable minimal hypersurfaces.
result Uniqueness of tangent cones with integer multiplicities.
Paper proves unique tangent maps for complex maps into algebraic varieties.
problem Proving uniqueness of tangent maps for complex maps into algebraic varieties.
method Developed techniques to prove uniqueness of tangent maps for weakly holomorphic and locally approximable maps.
result Established unique tangent cone property for weakly holomorphic maps into projective algebraic varieties.
Extends differential geometry concepts to manifolds with super tangent bundles.
problem No specific problem stated; extending differential geometry to super tangent bundles.
method Introduces super tangent bundle and extends differential geometry concepts.
result Basic notions of differential geometry extended to manifolds with super tangent bundles.
Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two tangents coincide with a given tangent, then we obtain the normal curvature of t…
Researchers compute the cohomology of an elliptic tangent bundle.
problem Computing the cohomology of a specific Lie algebroid.
method Direct computation of cohomology.
result The cohomology of the elliptic tangent bundle is computed.
We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex 3-folds of the form P(T∗X) whose tangent bundles are nef. Moreover, we show that if X is a Fano manifold such t…
The paper defines a new structure on tangent sphere bundles and characterizes their properties.
problem Characterizing properties of tangent sphere bundles with contact pseudo-metric structures.
method Introduced a contact pseudo-metric structure on TεM and proved manifold properties based on constant sectional curvature. result The tangent sphere bundle TεM is (κ,μ)-contact pseudo-metric manifold if and only if the manifold M has constant sectional curvature. New Calabi-Yau metrics found on complex symmetric spaces.
problem Finding Calabi-Yau metrics on complex symmetric spaces.
method Complete Calabi-Yau metrics with prescribed horospherical singular tangent cone.
result First examples of Calabi-Yau smoothings of singular tangent cones.
A novel method for parallel transport and geodesics on submanifolds.
problem Understanding parallel transport and geodesics on submanifolds.
method Rolling tangent space to visualize and analyze parallel transport and geodesics.
result Conditions for parallel transport and geodesics are simplified and visualized in the tangent space.
Study geodesics and F-geodesics on tangent bundles over para-Kähler-Norden manifolds.
problem Investigate geodesics and F-geodesics on tangent bundles.
method Investigate geodesics and F-geodesics on tangent bundles and φ-unit tangent bundles equipped with φ-Sasaki metric over para-Kähler-Norden manifolds.
result Investigate and analyze geodesics and F-geodesics on tangent bundles.
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an amb…
Characterizes special curves on surface tangent bundles.
problem Understanding curves on surface tangent bundles.
method Characterization of Legendre and slant curves.
result Characterizations for N-Legendre and N-slant curves.
We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having an affine structure on a manifold is equivalent to having a flat torsion-free c…
The paper studies singularities on parallels of tangent developable surfaces of frontal curves.
problem Understanding singularities on parallels of tangent developable surfaces.
method Generalization of tangent developable surfaces and parallel deformations for frontal curves in arbitrary dimensions.
result Classification of generic singularities on parallels of tangent developable surfaces for frontal curves in 3 or 4 dimensional Euclidean spaces.
Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying all invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Explicitly obtained all structures, distinguishing K-contact, Sasakian, and 3-Sasakian structures.
result There is a unique Sasakian-Einstein metric on tangent sphere bundles of spheres and real projective spaces.
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an amb…
The Samuelson condition is not satisfied by tangent lines of quadratic curves.
problem Area condition for Lagrangian 2-web
method Show that the Samuelson condition is not satisfied
result The Samuelson condition is not satisfied by tangent lines of quadratic curves.
Uniqueness proven for cylindrical tangent cones in high dimensions.
problem Proving uniqueness of cylindrical tangent cones in high dimensions.
method Analyzing area-minimizing hypersurfaces in R^9.
result Uniqueness of cylindrical tangent cones Cp,qimesR in R9. We explore the intrinsic geometry of tangent bundles and properties of the mirror map.
problem Understanding the intrinsic geometry of tangent bundles and properties of the mirror map.
method Review and solve cohomological questions related to the mirror map and related operators.
result Solved some cohomological questions and raised other induced by the d_B operator.
Study compact Kähler manifolds with pseudo-effective tangent bundles.
problem Characterize compact Kähler manifolds with strongly pseudo-effective tangent bundles.
method New proofs and characterizations of properties of vector bundles.
result Only projective spaces have big tangent bundles.
A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphis…
Study shows a universal local obstruction to the Samuelson condition for tangent Lagrangian 2-webs.
problem Obstruction to the Samuelson condition for tangent Lagrangian 2-webs.
method Local analysis of tangent lines and their intersection maps.
result A universal local phenomenon produces a nonzero mixed derivative, obstructing the Samuelson condition.
Study positive characteristic Fano 4-folds with nef tangent bundles.
problem Positive characteristic version of the Campana-Peternell conjecture for Fano 4-folds.
method Analyzes Fano 4-folds with nef tangent bundles in positive characteristic.
result Affirmative answer for Fano 4-folds with Picard number > 1 and nef tangent bundle.
We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth cur…
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
Unique cylindrical tangent cone for Simons' hypersurface found.
problem Uniqueness of cylindrical tangent cones for area-minimizing hypersurfaces.
method Developed a new Lojasiewicz inequality for non-isolated singularities.
result Cylindrical tangent cone for Simons' hypersurface is unique.
The article investigates conditions for isomorphism of singular tangent bundles.
problem Conditions for isomorphism of singular tangent bundles.
method Logarithmic and b-tangent bundles approach to resolve singularities. result Established a Poincaré-Hopf theorem for bm-tangent bundles. Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which all…
The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.
problem Understanding the structure of projective klt varieties with nef tangent sheaves.
method Developing theory of positivity of coherent sheaves and proving structure theorem.
result Projective klt varieties with specific tangent sheaf properties admit rationally connected fibrations onto abelian varieties.
We show, using two different approaches, that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian…
Steinhaus conjectured that every closed oriented C1-curve has a pair of anti-parallel tangents. Porter disproved the conjecture by showing that there exist curves with no anti-parallel tangents. Colin Adams rised the question of whether there exists a nontrivial knot in R3 which has no parallel or antiparallel t…
Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.
problem Constructing a Cartan calculus in tangent categories.
method Define scalar multiplication by a commutative ring object R to equip tangent bundles with R-module structure. result Every object in tangent categories carries a Cartan calculus of Lie-Rinehart forms.
The article calculates the F-convergence rate for Ricci flows with closed and smooth tangent flows.
problem Analyzing the convergence rate of Ricci flows with specific tangent flows.
method Calculating the F-convergence rate for Ricci flows with closed and smooth tangent flows. result A Ricci flow with closed and smooth tangent flow is ∣logλ∣−θ close to its tangent flow in the F-sense. We observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler-Lagrange equations on the 2nd order tangent group from the 1st order Euler-Lagrange equ…
The purpose of the present work is to study the complete and horizontal lifts of the metallic structure on tangent bundles with respect to almost product structure. We also establish fundamental formulae related to integrability and horizontal lifts of metallic structures on tangent bundles. Moreover, the study reveale…
We compute the fundamental groups of the complements of the family of real conic-line arrangements with up to two conics which are tangent to each other at two points, with an arbitrary number of tangent lines to both conics. All the resulting groups turn out to be big.
We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the …