arXiv research

The study finds conditions for a third rank Killing tensor field on a 2D Riemannian torus.

For generic torus-invariant metrics, eigenspaces are 2D and nodal sets are connected hypersurfaces.

Explains a 2D color exchange invariant correspondence to 3D linking numbers.

Computes a new metric quantity Y(M) for Riemannian 2d-manifolds.

This paper studies both the conductance and charge transport on 2D orbifolds in a strong magnetic field. We consider a family of Landau Hamiltonians on a complex, compact 2D orbifold $Y$ that are parametrised by the Jacobian torus $J(Y)$ of $Y$. We calculate the degree of the associated stable holomorphic spectral orbi…

Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.

Discretizes Hodge-Dirac operators on a torus.

We give, using an explicit expression obtained in [V. Jones, Ann. of Math. 126, 335 (1987)], a basic hypergeometric representation of the HOMFLY polynomial of $(n,m)$ torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result the $(m,n)\leftrightarrow (n,m)$ s…

Study on a specific type of Riemannian manifolds constructed from 2D space-forms.

Researchers study Killing superalgebras in 2D manifolds.

In this paper we classify maps from a torus phase space $X$ to $\mathcal{H}_n^*$, the space of $n \times n$, non-singular hermitian operators up to equivariant homotopy. The equivariance is with respect to a time-reversal involution on $X$ and an involution on $\mathcal{H}_n^*$ defining a certain symmetry class. Furthe…

Explicitly describes pluriclosed metrics on compact Lie groups.

Study of symmetries in a 2D space with specific metric properties.

A 2D Riemannian space has only 2 injective geodesics.

Example shows no global coordinates on 2-torus's cover.

New equations on manifolds linked to torus actions, proving existence and uniqueness.

This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.

Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.

Study nearly parallel G2-structures with torus symmetry using multi-moment maps.

New curvature K(x) measures manifold properties without integrals.

Paper finds new criteria for conjugate points in fluid flows.

Study on $(λ,λ)$-eigenfunctions on compact manifolds, showing manifold properties and eigenfamily dimensions.

We give an affirmative answer to the Halperin-Carlsson conjecture for the homologically injective torus actions on closed manifolds. This class contains holomorphic torus actions on compact Kahler manifolds, torus actions on compact Riemannian flat manifolds.

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional cu…

Classifies symplectic torus actions up to equivariant symplectomorphism.

Study shows how to deform foliated manifolds with flat leaves, leading to geometric characterization.

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank…

Proves torus sequences can't collapse to intervals under curvature bounds.

New K-theory approach classifies anyonic topological phases in 2D semimetals.

We point out that a 4-dimensional topological manifold with an Alexandrov metric (of curvature bounded below) and with an effective, isometric action of the circle or the 2-torus is locally smooth. This observation implies that the topological and equivariant classifications of compact, simply connected Riemannian 4-ma…

Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.

It is shown that in every dimension n=3j+2, j=1,2,3,..., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial tor…

We show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian $(n+2)$-manifold, with regular leaves homeomorphic to the $n$-torus, is given by a smooth effective $n$-torus action. This solves in the negative for the codimension $2$ case a question about the existence of foliat…

The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.

Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from TSigma to any Lie algebroid E, where Sigma is regarded as d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S^ij(X) A_…

We study fixed points of smooth torus actions on closed manifolds using fixed point formulas and equivariant elliptic genera. We also give applications to positively curved Riemannian manifolds with symmetry.

We answer a question of Liokumovich-Nabutovsky-Rotman showing that if D is a Riemannian 2-disc with boundary length L, diameter d and area A << d then D can be filled by a homotopy where the lengths of the intermediate curves are bounded by $L+2d+O(\sqrt A)$.

We construct a new example of an A-manifold, i.e. a Riemannian manifold with a cyclic-parallel Ricci tensor, which can be viewed as a generalization of the Einstein condition. The underlying manifold for our construction is a principal torus bundle over Kähler-Einstein manifold a with fibre a torus of arbitrary dimensi…

In this paper we prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure ass…

Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.

We obtain an equivariant classification for orientable, closed, four-dimensional Alexandrov spaces admitting an isometric torus action. This generalizes the equivariant classification of Orlik and Raymond of closed four-dimensional manifolds with torus actions. Moreover, we show that such Alexandrov spaces are equivari…

Quantization on even-dimensional compact manifolds using cell decomposition.

Explains examples of Lagrangian flow with circle symmetry.

A symmetric tensor field on a Riemannian manifold is called Killing field if the symmetric part of its covariant derivative is equal to zero. There is a one to one correspondence between Killing tensor fields and first integrals of the geodesic flow which depend polynomially on the velocity. Therefore Killing tensor fi…

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if $T$ is a Riemannian 2-torus with boundary in $\mathbb R ^n$, such that the boundary curve is a standard unit circle, then the length o…

The paper classifies manifolds with free torus actions and positive Ricci curvature.

The paper tackles scalar curvature on manifolds conformal to tori, proving stability under certain conditions.

The study connects matroids with torus representations and positive curvature.