Curves with constant torsion can be deformed arbitrarily.
problem Deforming curves of constant torsion in Euclidean space.
method Convex integration and degree theory.
result Existence of knots with constant torsion in each isotopy class.
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48 results for “constant torsion”
Ruled surfaces with Ricci metrics use curves of constant torsion.
problem Characterizing ruled surfaces with Ricci metrics.
method Using curves of constant torsion to construct ruled surfaces.
result Helicoid is the only surface with constant mean curvature.
We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.
Study of curves in dual space with constant curvature and torsion.
problem Classifying curves in dual space with specific geometric properties.
method Defined curvature and torsion for curves in dual space, classified curves with constant properties, and proved existence theorems.
result Established fundamental theorem of existence for dual curves with prescribed curvature and torsion.
Study path geometries with constant torsion and cone structures.
problem Characterizing path geometries with nontrivial torsion.
method Introducing constant torsion, establishing correspondence with cone structures, describing in terms of integrable systems.
result Path geometries with constant torsion correspond to cone structures on homogeneous ruled surfaces.
We prove that the mod Z reduction of the torsion of a rational homology 3-sphere is completely determined by three data: a certain canonical spin^c structure, the linking form and a Q/Z-valued constant c. This constant is a new topological invariant of the rational homology sphere. Experimentations with lens spaces sug…
The paper confirms a conjecture for Bismut torsion parallel metrics.
problem The existence of metrics with constant holomorphic sectional curvature on non-Kähler manifolds.
method Investigation of Bismut torsion parallel metrics.
result The conjecture is confirmed for all non-balanced BTP manifolds.
Salkowski \cite{salkow}, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\e^3$. In this paper, we adapt definition of such curves to ti…
Study geometric flow on curves with positive torsion, finding stationary solutions and their stability.
problem Analyzing geometric flow on curves with positive torsion.
method Evolution equation , studying stationary solutions and linear stability.
result Explicit formula for stationary solutions of helices with constant curvature and torsion, proving stability.
The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vo…
Study on triharmonic curves in Sol space with constant curvature and torsion.
problem Characterizing triharmonic curves in the Sol space.
method Complete classification of proper triharmonic curves with constant geodesic curvature and torsion.
result Triharmonic curves form a constant angle with a Killing field of constant length.
We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded…
The study explores properties of metric connections with skew torsion and their curvature identities.
problem Investigating curvature properties of metric connections with skew torsion.
method Analyzing the curvature and torsion properties of metric connections with skew torsion.
result Necessary and sufficient conditions for a metric connection with skew torsion to satisfy the Riemannian first and second Bianchi identities are presented.
The paper derives formulas for linear connections with totally anti-symmetric torsion in 3D generalized Berwald manifolds.
problem Understanding linear connections with specific torsion in generalized Berwald manifolds.
method Averaging of Finslerian quantities over indicatrix surfaces to express the torsion tensor.
result Explicit formulas for linear connections with totally anti-symmetric torsion in 3D generalized Berwald manifolds.
Study curvature properties of connections with skew-symmetric torsion.
problem Investigate curvature identities and solitons on manifolds.
method Analyzes curvature identities and properties of connections with skew-symmetric torsion.
result Characterizes conditions for curvature to be symmetric and Ricci flat.
The paper defines and analyzes the adjoint Reidemeister torsion for connected sums of knots.
problem Defining and analyzing the adjoint Reidemeister torsion for connected sums of knots.
method Defined a natural way to compute the adjoint Reidemeister torsion for high-dimensional components of the character variety.
result The adjoint Reidemeister torsion is locally constant and satisfies the vanishing identity.
The paper explores algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
problem Investigating algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
method Analyzing twisted Alexander polynomials and Reidemeister torsions of torus knots associated with irreducible -representations.
result Proves that coefficients of twisted Alexander polynomials are locally constant functions on the -character variety.
Analytic torsion defined for rank 2 distributions on 5-manifolds.
problem Defining and analyzing analytic torsion for rank 2 distributions.
method Proposed an analytic torsion for Rumin complex associated with rank 2 distributions on 5-manifolds, established anomaly formulas, and showed coincidence with Ray-Singer torsion.
result The proposed torsion coincides with Ray-Singer torsion for certain nilmanifolds.
The curvature properties of a specific type of 6-manifold are explored.
problem Curvature identities on almost Calabi-Yau 6-manifolds with torsion.
method Analysis of the Nijenhuis tensor, torsion connection, and Ricci solitons.
result The curvature of the torsion connection is symmetric and vanishes if the norm of the torsion or scalar curvature is constant.
In this paper we define the Reidemeister torsion as a rational function on the geometric components of the character variety of a one-cusped hyperbolic manifold M. We study its poles and zeros, and we deduce sufficient conditions on the manifold M for this function being non-constant.
Paper studies how discrete space curves with constant torsion deform to model linkage motions.
problem Modeling and understanding the motion of discrete space curves with constant torsion.
method Using semi-discrete mKdV equations to describe the motion of discrete space curves.
result The motion of discrete space curves is governed by semi-discrete mKdV equations.
Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the sequence of torsion numbers satisfies a linear homogeneous recurrence relation …
1-loop invariant equals torsion for 2-bridge knots.
problem Proving a conjecture about knot invariants.
method Combining Ohtsuki-Takata work with explicit computation.
result 1-loop term equals Reidemeister torsion for hyperbolic 2-bridge knots.
Guts determine the leading coefficients of -Alexander torsions for 3-manifolds.
problem Determining the leading coefficient of -Alexander torsions for 3-manifolds.
method Using a new criterion for the convergence of Fuglede-Kadison determinants and the work of Agol and Zhang on guts of 3-manifolds.
result The leading coefficient equals the relative -torsion of the guts associated to the cohomology class.
This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection with skew symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is …
We prove that a Kleinian group acting upon admits a non-constant -automorphic function, even if it has torsion elements, provided that the orders of the elliptic (torsion) elements are uniformly bounded. This is accomplished by developing a technique for mashing distinct fat triangulations while…
New dynamical torsion for contact Anosov flows connects to Reidemeister torsion.
problem Understanding contact Anosov flows and their properties.
method Introducing dynamical torsion and showing its properties.
result Locally constant ratio between dynamical and Turaev torsion.
Explicit relation found between knot torsion and TQFT signatures.
problem Relating Reidemeister torsion of two-bridge knots to TQFT signatures.
method Established an explicit relation using parabolic representations and SU-TQFT.
result Inverse sum of torsions is constant and signatures have similar asymptotic behavior.
The purpose of this article is to give an explicit formula for all curves of constant torsion in the unit two-sphere . These curves and their basic properties have been known since the 1890's, and some of these properties are discussed in the Appendix. Some example curves, computed with a standard ODE packa…
Characterizes curves in totally umbilical surfaces of space forms.
problem Identifying curves in totally umbilical surfaces of space forms.
method Analyzes curvature and torsion conditions for curves in $\h^3$ and $\s^3$.
result Necessary and sufficient conditions for curves in totally umbilical surfaces.
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.
The main topic of this paper is to show that in the 3-dimensional Minkowski spacetime, the torsion of a null curve is equal to the Schwarzian derivative of a certain function appearing in a description of the curve. As applications, we obtain descriptions of the slant helices, and null curves for which the torsion is o…
The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under the assumption of the product structure near boundary. As applications of this result,we prove the adiabatic decomposi…
We provide new results and new proofs of results about the torsion of curves in . Let be a smooth curve in that is the graph over a simple closed curve in with positive curvature. We give a new proof that if has nonnegative (or nonpositive) torsion, then has zero …
The paper confirms a conjecture about Hermitian manifolds with constant mixed curvature.
problem Compact Hermitian manifolds with non-zero constant mixed curvature must be Kähler.
method Verification for specific types of Hermitian manifolds including complex nilmanifolds, solvmanifolds, and Lie algebras.
result Partial evidence supporting Kai Tang's conjecture.
Study natural and conjugate mates of Frenet curves in Lie groups.
problem Characterize Frenet curves and their mates in Lie groups.
method Introduced natural and conjugate mates, derived relationships, analyzed specific curves.
result Obtained results for various Frenet curves in Lie groups.
Post-Lie algebra structure found on non-flat manifolds with curvature and torsion.
problem Understanding vector fields and endomorphisms on manifolds with curvature and torsion.
method Analyzing the post-Lie algebra structure of vector fields and endomorphisms for non-flat connections.
result A universal Lie algebra is constructed for the post-Lie algebra of vector fields and endomorphisms.
The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also holds for connections with metric torsion in the case of non-positive Euler characte…
We study special almost Kaehler manifolds whose curvature tensor satisfies the second curvature condition of Gray. It is shown that for such manifolds, the torsion of the first canonical Hermitian is parallel. This enables us to show that every AK_2-manifold has parallel torsion. Some applications of this result, conce…
The paper studies curvature identities and solitons on Spin(7)-manifolds.
problem Curvature identities and solitons on Spin(7)-manifolds.
method Analyzes the curvature and torsion of Spin(7)-manifolds, proving identities and conditions.
result Conditions for closed torsion and implications for Ricci flatness and solitons.
We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We…
On the curvature of the Bismut connection: Bismut Yamabe problem and Calabi-Yau with torsion metricsmath.DG
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
New Lehmer constants computed for free groups, improving bounds.
problem Tackles Lehmer's problem on Mahler measures and Fuglede-Kadison determinants.
method Uses relations between Fuglede-Kadison determinants and random walks on Cayley graphs, and gluing formula for -torsions.
result Computes new upper bound of for Lehmer's constants of torsion-free groups with non-cyclic free subgroups.
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in se…
Suppose is a compact connected odd-dimensional manifold with boundary, whose interior comes with a complete hyperbolic metric of finite volume. We will show that the -topological torsion of and the -analytic torsion of the Riemannian manifold are equal. In particular, the -top…
Study the geometry and symmetries of moduli spaces of connections on KT manifolds.
problem Investigate the geometry and symmetries of moduli spaces of Hermitian-Einstein and instanton connections on KT manifolds.
method Analyze the geometry and symmetries of moduli spaces of Hermitian-Einstein and instanton connections on KT manifolds, using vector fields and connections with skew-symmetric torsion.
result The geometry of moduli spaces can be modeled on principal bundles with specific fibre and base spaces.
We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any non-degenerated 2-form or any spinor. Suitable integral formulas allow us to prove similar properties in case …
The moduli space of Hermitian-Einstein connections on certain manifolds has a strong Kähler with torsion structure.
problem Characterizing the moduli space of Hermitian-Einstein connections on manifolds with a dilaton field.
method Demonstrates the existence of a strong Kähler with torsion structure on the moduli space under specific conditions on the Lee form and dilaton field.
result The moduli space admits an induced holomorphic and Killing vector field when the manifold has a holomorphic and Killing vector field invariant under the dilaton.