arXiv research

Study classifies 2-manifolds with special homeomorphism groups.

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinator…

We introduce the notion of complex $G_2$ manifold $M_{\mathbb C}$, and complexification of a $G_2$ manifold $M\subset M_{\mathbb C}$. As an application we show the following: If $(Y,s)$ is a closed oriented $3$-manifold with a $Spin^{c}$ structure, and $(Y,s)\subset (M, \varphi)$ is an imbedding as an associative subma…

The study provides homological characterizations for $Q$-manifolds and $l_2$-manifolds.

We define deformations of $RP^2$-manifolds.

Game theory applied to splitting surfaces of compact 2-manifolds.

Conditions for Penrose-Ward transformation on specific manifolds.

The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.

The paper constructs complex hyperbolic 2-manifolds with one cusp.

Formula derived for $G_2$-manifolds, showing moduli spaces are incomplete.

In this survey, we describe invariants that can be used to distinguish connected components of the moduli space of holonomy G_2 metrics on a closed 7-manifold, or to distinguish G_2-manifolds that are homeomorphic but not diffeomorphic. We also describe the twisted connected sum and extra-twisted connected sum construc…

We consider the minimum Yang-Mills energy on the complete $G_{2}$-manifolds and Calabi-Yau 3-folds,the connection $A$ is a stability Yang-Mills connection on the $G$-bundle $E$.We prove that the connection must be a $G_{2}$-instanton on $G_{2}$-manifold and the bundle is holomorphic on Calabi-Yau 3-fold with holonomy $…

Classifies manifolds with dense conjugacy classes in their mapping class groups.

Study constructs associative submanifolds in $G_2$-manifolds from orbifolds.

We prove that the homeomorphism problem for 2-manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.

This is a very short and elementary introduction to G_2 manifolds. We stress the similarities and the differences with Kähler manifolds in general and with Calabi-Yau manifolds in particular.

On a projective complex manifold, the Abelian group of Divisors maps surjectively onto that of holomorphic line bundles (the Picard group). On a $G_2$-manifold we use coassociative submanifolds to define an analogue of the first, and a gauge theoretical equation for a connection on a gerbe to define an analogue of the …

Study of $U(1)$ gauge theories on $G_2$ manifolds, including instantons and Chern-Simons.

M-theory compactified on $G_2$-holonomy manifolds results in 4d $\mathcal{N}=1$ supersymmetric gauge theories coupled to gravity. In this paper we focus on the gauge sector of such compactifications by studying the Higgs bundle obtained from a partially twisted 7d super Yang-Mills theory on a supersymmetric three-cycle…

Study geometry on $G_2$-manifolds, focusing on $Spin(7)$ and $G_2$.

Study $G_2$-manifolds from symplectic $SU(3)$-manifolds with $T^2$-symmetry.

Computes a $ν$-invariant for Joyce's $G_2$-manifolds.

Article constructs coassociative submanifolds in Joyce's $G_2$-manifolds.

We obtained that any 2-form and any smooth function on 2-manifolds with boundary can be realized as the curvature form and the gaussian curvature function of some Riemmanian metric, respectively.

The paper constructs diffeomorphisms on a $G_2$-manifold achieving entropy bounds.

We construct the M-Theory lifts of type IIA orientifolds based on K3-fibred Calabi-Yau threefolds with compatible involutions. Such orientifolds are shown to lift to M-Theory on twisted connected sum $G_2$ manifolds. Beautifully, the two building blocks forming the $G_2$ manifold correspond to the open and closed strin…

In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of co…

For an arbitrary nondegenerate curve in a pseudo-Riemann\-ian (including Riemannian) 2-manifold, we express the equi-affine curvature with the help of the Frenet (geodesic) curvature of this curve.

We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.

We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.

The study constructs $G_2$-manifolds from K3 surfaces with a specific action.

We study supergravity solutions corresponding to fivebranes wrapped on a three-sphere inside a G_2 holonomy manifold. By changing a parameter the solutions interpolate between a G_2 manifold X_i \cong S^3 x R^4 with flux on a three-sphere and a distinct G_2 manifold X_j \cong S^3 x R^4 with branes on another three-sphe…

We give a full description of totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold of constant curvature and present a new class of a cylinder-type totally geodesic submanifolds in the general case.

Study on deformation theory of nearly G2 manifolds with obstructions.

Explicit formulas for the $G_2$-components of the Riemannian curvature tensor on a manifold with a $G_2$ structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact $G_2$ manifol…

Compact non-formal $G_2$ manifold with $b_1=1$.

In this article, we study an almost contact metric structure on a $G_2$-manifold constructed by Arikan, Cho and Salur in via the classification of almost contact metric structures given by Chinea and Gonzalez. In particular, we characterize when this almost contact metric structure is cosymplectic and narrow down the p…

Method counts connected 2D stratifolds with singular curves and components.

We demonstrate how by using the intersection theory to calculate the cohomology of $G_2$-manifolds constructed by using the generalized Kummer construction. For one example we find the generators of the rational cohomology ring and describe the product structure.

For two nearby disjoint coassociative submanifolds C and C' in a G_2-manifold, we construct thin instantons with boundaries lying on C and C' from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.

We argue that G_2 manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M5-brane fibers as target space. Mirror symmetry and various string and M-theory dualities involving G_2 manifolds may be incorporated into…

In this paper we study a Hamiltonian function on the cotangent bundle of the space of Riemannian metrics on a 3-manifold $M$ and prove the orbits of the constrained Hamiltonian dynamical system correspond to $G_2$-manifolds foliated by hypersurfaces diffeomorphic to $M\times \mathrm{SO}(3)$.

Any 7-dimensional cocalibrated G_2-manifold admits a unique connection $\nabla$ with skew symmetric torsion. We study these manifolds under the additional condition that the $\nabla$-Ricci tensor vanishes. In particular, we describe their geometry in case of a maximal number of $\nabla$-parallel vector fields.

Study $G_2$ manifolds with symmetry, finding new submanifolds.

Here we study the deformations of associative submanifolds inside a G_2 manifold M^7 with a calibration 3-form φ. A choice of 2-plane field Λon M (which always exits) splits the tangent bundle of M as a direct sum of a 3-dimensional associate bundle and a complex 4-plane bundle TM= E\oplus V, and this helps us to relat…

We construct examples of exponentially asymptotically cylindrical Riemannian 7-manifolds with holonomy group equal to G_2. To our knowledge, these are the first such examples. We also obtain exponentially asymptotically cylindrical coassociative calibrated submanifolds. Finally, we apply our results to show that one of…