arXiv research

New metrics produce discrete zero sets for nondegenerate harmonic forms.

Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.

GN algorithm solves batched bandit for nondegenerate functions near-optimally.

Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.

Minimal surfaces in 8D smooth and nondegenerate.

Models of 2-nondegenerate CR hypersurfaces in C^N are characterized and their defining equations simplified.

In Euclidean geometry, all metric notions (arc length for curves, the first fundamental form for surfaces, etc.) are derived from the Euclidean inner product on tangent vectors, and this inner product is preserved by the full symmetry group of Euclidean space (translations, rotations, and reflections). In equiaffine ge…

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.

In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie gr…

The purpose of this work is to extend the formalism of stochastic calculus to the case of spaces with local anisotropy (modeled as vector bundles with compatible nonlinear and distinguished connections and metric structures and containing as particular cases different variants of Kaluza--Klein and generalized Lagrange …

We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=\frac{1}{2}£_ξ\varphi$ and $\ell := R(\cdot,ξ)ξ$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $ξ$-sectional curvatures are obtained. We establish necessary and su…

Paper shows pseudometrics on braid groups are nondegenerate.

A smooth fibration of $\mathbb{R}^3$ by oriented lines is given by a smooth unit vector field $V$ on $\mathbb{R}^3$, for which all of the integral curves are oriented lines. Such a fibration is called skew if no two fibers are parallel, and it is called nondegenerate if $\nabla V$ vanishes only in the direction of $V$.…

New Einstein metric found on non-standard solvmanifold.

We formulate a Calabi-Yau type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kähler structures generalizing the notion of Kähler class, we conjecture unique solvability of Gualtieri's Calabi-Yau e…

We study CR hypersurfaces in C^4 with constant rank Levi form and find their defining equations.

We prove that any two Kahler potentials on a compact Kahler manifold can be connected by a geodesic segment of C^{1,1} regularity. This follows from an a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampere equation, which is independent of a positive lower bound for the right hand…

The study identifies two sources of invariants in 2--nondegenerate CR geometries.

We define a nondegenerate Monge-Ampère structure on a 6-dimensional manifold as a pair $(Ω,ω)$, such that $Ω$ is a symplectic form and $ω$ is a 3-differential form which satisfies $ω\wedgeΩ=0$ and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi-Yau stru…

Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map Π: \wedge^2 W \to V. If the skew symmetric vector valued bilinear form Πis nondegenerate then (M,Q) i…

The paper explores CR structures and their leaf spaces in semi-Riemannian manifolds.

In this article, we solve the equivalence problem for 2--nondegenerate CR geometries that have (at every point) a homogeneous space $G/H$ as a maximally symmetric model for $G$ simple real Lie group of CR automorphisms. This completes the classification of real submanifolds in complex space that are maximally symmetric…

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the volume-preserving diffeomorphism group of a three-dimensional manifold and study its properties. Despite the fact that the space $\mathcal{D}_μ(M^3)$ is infinite-dimensional, we succeed in defining the signature of the bi-invariant quadr…

Defines metric bundles for manifold geometries, unifying various types of metrics.

A parametric curve $γ$ of class $C^n$ on the $n$-sphere is said to be nondegenerate (or locally convex) when $\det\left(γ(t),γ'(t),\cdots,γ^{(n)}(t)\right)>0$ for all values of the parameter $t$. We orthogonalize this ordered basis to obtain the Frenet frame $\mathfrak{F}_γ$ of $γ$ assuming values in the orthogonal gro…

For an almost complex structure $J$ in dimension 6 with nondegenerate Nijenhuis tensor $N_J$, the automorphism group $G=Aut(J)$ of maximal dimension is the exceptional Lie group $G_2$. In this paper we establish that the sub-maximal dimension of automorphism groups of almost complex structures with nondegenerate $N_J$,…

Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-o…

We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold $M$, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the $C^k$-topology, $k=2,...,\infty$, in the set of metrics of …

We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surface…

The information metric arises in statistics as a natural inner product on a space of probability distributions. In general this inner product is positive semi-definite but is potentially degenerate. By associating to an instanton its energy density, we can examine the information metric {\bf g} on the moduli spaces $\M…

Researchers create normal forms for CR manifolds in complex space.

We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a b…

We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal D,\mathcal J)$ for all $k\geq 2$ and call this invariant the core …

3D contact forms have supporting decompositions, leading to entropy results.

The study finds nondegenerate harmonic 1-forms using symmetry conditions.

Study CR manifolds focusing on Levi and contact-nondegeneracy.

This article proves that if M is a smooth manifold of dimension at least four, then for generic choice of metric on M, all prime parametrized minimal surfaces in M are free of branch points and lie on nondegenerate critical submanifolds for the two-variable energy function which have the same dimension as the group of …

New method classifies symplectic structures on Lie groups.

In curved spaces, isoperimetric sets don't exist for small volumes.

Study metric perturbations to make degenerate harmonic forms non-degenerate.

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

Generic smooth boundaries for isoperimetric regions in 8D manifolds.

New Lie groups generalize H-type groups with nondegenerate centers.

Characterizes CR manifolds in complex flag manifolds.

Constructs metrics with Q-curvature on manifolds with singularities.

We show that a compact Kahler manifold admitting a nondegenerate holomorphic 2-form valued in a line bundle is a finite cyclic cover of a hyperkahler manifold. With respect to the connection induced by the locally hyperkahler metric, the form is parallel. We then describe the structure of the fundamenal group of such m…

Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.

Let $V$ be a finite dimensional vector space over a field $F$ of characteristic different from 2, and let $Q$ be a nondegenerate, symmetric, bilinear form on $V$. Let $C\ell(V,Q)$ be the Clifford algebra determined by $V$ and $Q$. The bilinear form $Q$ extends in a natural way to a nondegenerate, symmetric, bilinear fo…