arXiv research

Introduces $\mathbb{K}$-framings for surfaces, generalizing quadratic forms.

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$, and $\mathbb{F}^1$ the space of framed Morse functions, both endowed with $C^\infty$-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping $\mathbb{F}^0\hookright…

Harmonic maps link Teichmüller spaces to framed representations.

In this study, we defined Fermi-Walker derivative in Galilean space $\mathbb{G}^3$. Fermi-Walker transport and non-rotating frame by using Fermi- Walker derivative are given in $\mathbb{G}^3$. Being conditions of Fermi-Walker transport and non-rotating frame are investigated along any curve for Frenet frame and Darboux…

Study of generalized Bishop frames on time-like curves in 4D Lorentz space.

Let $\mathbb F$ be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in ${\rm Sp}(2n,\mathbb F)$. We show that ultralimits of maximal representations in ${\rm Sp}(2n,\mathbb R)$ admit such a framing, and that all maximal framed represen…

Study of generalized Bishop frames on curves in 4D space.

The paper connects hyperbolic spinors to non-null framed curves in Minkowski 3-space.

In this study, we consider canal surfaces according to parallel transport frame in Euclidean space $\mathbb{E}^{4}$. The curvature properties of these surfaces are investigated with respect to $k_{1}$, $k_{2}$ and $k_{3}$ which are principal curvature functions according to parallel transport frame. We also give an exa…

We introduce an Alexander polynomial for MOY graphs. For a framed trivalent MOY graph $\mathbb{G}$, we refine the construction and obtain a framed ambient isotopy invariant $Δ_{(\mathbb{G},c)}(t)$. The invariant $Δ_{(\mathbb{G}, c)}(t)$ satisfies a series of relations, which we call MOY-type relations, and conversely t…

The study shows ergodicity of unitary frame flows on Kähler manifolds with specific curvature conditions.

Extended RDS filtering for positions and orientations, improving crossing structure enhancement and inpainting.

Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the sing…

Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.

New method for curve comparison using iterated integrals and moving frames.

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $α:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(α(x))_k=|<x,f_k>|^2$, where $…

This thesis is devoted to the Differential Geometry of curves and surfaces along with applications in Quantum Mechanics. In the 1st part we introduce the well known Frenet frame. Later, we show that the curvature function is a lower bound for the scalar angular velocity of any other orthonormal moving frame, from which…

We consider an $n$-dimensional projective space $\mathbb{P}_n$ ($n\geq2$) and a fixed point $A$ on it. Let $F(\mathbb{P}_n)$ be the manifold of all the projective frames of $\mathbb{P}_n$ having $A$ as their first vertice. We define the action of stabilizer G of $A$ in the projective group $GP(n)$ in a natural way. The…

The paper studies connectivity of Schur-Horn map images in real Grassmannians.

Study calculates instanton Floer homology for surgeries on L-space knots.

Study contact 3-manifolds with special frames, deriving curvature bounds.

Characterizes visibility and geodesic loops in complex domains.

Paper studies rigidity of translation surfaces in 3D sphere using quaternionic product.

Locally stable maps $S^3\to\mathbb{R}^4$ are classified up to homotopy through locally stable maps. The equivalence class of a map $f$ is determined by three invariants: the isotopy class $σ(f)$ of its framed singularity link, the generalized normal degree $ν(f)$, and the algebraic number of cusps $κ(f)$ of any extensi…

The paper shows that random frames have full spark with high probability.

New method constructs relative invariants for group actions on extended manifolds.

We study the open string integrality invariants (LMOV invariants) for toric Calabi-Yau 3-folds with Aganagic-Vafa brane (AV-brane). In this paper, we focus on the case of the resolved conifold with one out AV-brane in any integer framing $τ$, which is the large $N$ duality of the Chern-Simons theory for a framed unknot…

We introduce BV-algebra structures on the homology of the space of framed long knots in $\mathbb{R}^n$ in two ways. The first one is given in a similar fashion to Chas-Sullivan's string topology. The second one is defined on the Hochschild homology associated with a cyclic, multiplicative operad of graded modules. The …

The paper classifies time-like surfaces in a static space-time.

Study on framed surfaces with bounds on Morse index.

We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into $\mathbb{R}^{36}$ (resp. $\mathbb{R}^{83}$).

In this paper we establish the equivalence of solutions between Schrödinger map into $\mathbb{S}^2$ or $\mathbb{H}^2$ and their associated gauge invariant Schrödinger equations. We also establish the existence of global weak solutions into $\mathbb{H}^2$ in two space dimensions. We extend these ideas for maps into com…

Inspired by the properties of an $n$-frame of gradients $(\nabla f_1, \ldots, \nabla f_n)$ of a Morin map $f:M\rightarrow\mathbb{R}^n$, with $\dim M\geq n$, we introduce the notion of Morin singularities in the context of singular $n$-coframes and singular $n$-frames. We also study the singularities of generic 1-forms …

The Liouville symplectic form connects various moduli spaces in algebraic geometry.

We use Cartan's method of moving frames to compute a complete set of local invariants for nondegenerate, 2-dimensional centroaffine surfaces in $\mathbb{R}^5 \setminus \{0\}$ with nondegenerate centroaffine metric. We then give a complete classification of all homogeneous centroaffine surfaces in this class.

Study instanton homology and its connection to Froyshov's invariant.

In this paper, we characterize Riemannian 4-manifold in terms of its almost Hermitian twistor spaces $(Z,g_t,\mathbb{J}_{\pm})$. Some special metric conditions (including Balanced metric condition, first Gauduchon metric condition) on $(Z,g_t,\mathbb{J}_{\pm})$ are studied. For the first Chern form of a natural unitary…

Extends RDS filtering to position-orientation space for better image processing.

We prove that the quasi-homogenous symbols on the projective space $\mathbb{P}^n(\mathbb{C})$ yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball $\mathbb{B}^n$. These algebras are Banach but not $C^*$. We prove the existen…

We extend knot contact homology to a theory over the ring $\mathbb{Z}[λ^{\pm 1},μ^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be generalized to knots in arbitrary manifolds, distinguishes the unknot and can distinguish…

In this paper, it is shown that every closed hyperbolic 3-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two fram…

We extend the notion of $r$-minimality of a submanifold in arbitrary codimension to $u$-minimality for a multi-index $u\in\mathbb{N}^q$, where $q$ is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with…

We investigate the geometric properties of hyperbolic affine flat, affine minimal surfaces in the equiaffine space $\mathbb{A}^3$. We use Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of such surfaces and…

Let $A\neq 0$ be a complex number with $|A|\neq 1$. Let $M$ be a compact smooth oriented $3$-manifold, the $SU(3)$-skein space of $M$, $S_A(M)$, is the vector space over $\mathbb{C}$ generated by framed oriented links (including framed oriented trivalent graphs in $M$) quotient by the $SU(3)$-skein relations due to Ku…

Frame flows on certain symmetric spaces mix exponentially.

In this paper we classify the homotopy classes of proper maps $E\rightarrow \mathbb R^k$, where $E$ is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps $\mathbb R^n\rightarrow \mathbb R^k$. We find a stability range of such maps. We conclude with some remarks…

We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over $\mathbb{Q}$, $\mathbb{Z}$ and $\mathbb{Z}_2$, the Kervaire invariant, the $f$-invariant, and the $ν$-invariant. This requires either a framing or a corner structure. The canonical framing pro…

Complete normal forms for specific real hypersurfaces in complex space are constructed.