arXiv subjects

Arash Bazdar

Publications and source records attributed to Arash Bazdar.

5 recordsLinked to original sources

arXiv2020

Infinitesimal homogeneity and bundles

Let $Q\to M$ be a principal $G$-bundle, and $B_0$ a connection on $Q$. We introduce an infinitesimal homogeneity condition for sections in an associated vector bundle $Q\times_GV$ with respect to $B_0$, and, inspired by the well known Ambrose-Singer theorem, we prove the existence of a connection which satisfies a system of parallelism conditions. We explain how this general theorem can be used to prove the known Ambrose-Singer type theorems by an appropriate choice of the initial system of data.We also obtain new applications, which cannot be obtained using the known formalisms, e.g. a classification theorem for locally homogeneous spinors. Finally we introduce natural local homogeneity and local symmetry conditions for triples $(g,P\stackrel{p}{\to} M,A)$ consisting of a Riemannian metric on $M$, a principal bundle on $M$, and a connection on $P$. Our main results concern locally homogeneous and locally symmetric triples, and they can be viewed as bundle versions of the Ambrose-Singer and Cartan theorem.

math.DG↗

arXiv2020

Locally homogeneous connections on principal bundles over hyperbolic Riemann surfaces

Let $g$ be locally homogeneous (LH) Riemannian metric on a differentiable compact manifold $M$, and $K$ be a compact Lie group endowed with an $\mathrm {ad}$-invariant inner product on its Lie algebra $\mathfrak{k}$. A connection $A$ on a principal $K$-bundle $p:P\to M$ on $M$ is locally homogeneous if for any two points $x_1$, $x_2\in M$ there exists an isometry $φ:U_1\to U_2$ between open neighborhoods $U_i\ni x_i$ which sends $x_1$ to $x_2$ and admits a $φ$-covering bundle isomorphism preserving the connection $A$. This condition is invariant under the action of the automorphism group (gauge group) of the bundle, so the classification problem for LH connections leads to an interesting moduli problem: for fixed objects $(M,g,K)$ as above describe geometrically the moduli space of all LH connections on principal $K$-bundles on $M$ (up to bundle isomorphisms). Note that if $A$ is LH, then the associated connection metric $g_A$ on $P$ is locally homogenous, so it defines a geometric structure (in the sense of Thurston) on the total space of the bundle. Therefore this moduli problem is related to the classification of LH (geometric) Riemannian manifolds which admit a Riemannian submersion onto the given manifold $M$. Omitting the details, our moduli problem concerns the classification of geometric fibre bundles over a given geometric base. We develop a general method for describing moduli spaces of LH connections on a given base. Using our method we give explicit descriptions of these moduli spaces when the base manifold is a hyperbolic Riemann surface $(M,g)$ and $K\in\{S^1,PU(2)\}$. The case $K=S^1$ leads to a new construction of the moduli spaces of Yang-Mills $S^1$-connections on hyperbolic Riemann surfaces, and the case $K=PU(2)$ leads to a one-parameter family of compact, 5-dimensional geometric manifolds, which we study in detail.

math.DG↗

arXiv2018

Infinitesimal isometries of connection metric and generalized moment map equation

Let $(M,g)$ be a smooth Riemannian manifold, $K$ a compact Lie group and $p:P\to M$ a principal $K$-bundle over $M$ endowed with a connection $A$. Fixing a bi invariant inner product on Lie algebra $\mathfrak{k}$ of $K$, the connection $A$ and metric $g$ define a Riemannian metric $g_A$ on $P$. Let $\tilde {X}$ be the horizontal lift of vector field $X$ on $M$ and, let $ξ^ν$ be the vertical field associated with section $ν\in A^0(\mathrm{ad}( P))$ of the adjoint bundle. It is proved that the connection $A$ is invariant under the 1-parameter group of local diffeomorphism generated by $\tilde{ X}+ξ^ν$ if and only if $X$ and $ν$ satisfy the generalized moment map equation $ι_XF_A=-\nabla^Aν$. The Lie algebra of fiber preserving Killing fields of $(P,g_A)$ is studied, in the case where $K$ is compact, connected and semisimple.

math.DG↗

arXiv2018

The Ricci flow for circle bundles over surfaces

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

math.DG↗

arXiv2017

Locally homogeneous triples. Extension theorems for parallel sections and parallel bundle isomorphisms

Let $M$ be a differentiable manifold and $K$ a Lie group. A locally homogeneous triple with structure group $K$ on $M$ is a triple $(g, P\stackrel{p}{\to} M,A)$, where $p:P\to M$ is a principal $K$-bundle on $M$, $g$ is Riemannian metric on $M$, and $A$ is connection on $P$ such that the following locally homogeneity condition is satisfied: for every two points $x$, $x'\in M$ there exists an isometry $φ:U\to U'$ between open neighborhoods $U\ni x$, $U'\ni x'$ with $φ(x)=x'$, and a $φ$-covering bundle isomorphism $Φ:P_U\to P_{U'}$ such that $Φ^*(A_{U'})=A_U$. If $(g,P\stackrel{p}{\to} M,A)$ is a locally homogeneous triple on $M$, one can endow the total space $P$ with a locally homogeneous Riemannian metric such that $p$ becomes a Riemannian submersion and $K$ acts by isometries. Therefore the classification of locally homogeneous triples on a given manifold $M$ is an important problem: it gives an interesting class of geometric manifolds which are fibre bundles over $M$. In this article we will prove a classification theorem for locally homogeneous triples. We will use this result in a future article in order to describe explicitly moduli spaces of locally homogeneous triples on Riemann surfaces.

math.DG↗