arXiv subjects

Udhav Fowdar

Publications and source records attributed to Udhav Fowdar.

14 recordsLinked to original sources

arXiv2026

Sasaki with torsion manifolds and string backgrounds

Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a $\nabla$-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact $\nabla$-Einstein manifold in dimension $5$ and $7$. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with $S^1$.

math.DG↗

arXiv2026

Flows of geometric structures II

We advance the general theory of flows of tensorial $\mathrm{H}$-structures, focusing on non-isometric flows and on the case $\mathrm{H}=\mathrm{SU}(m)\subset\mathrm{SO}(2m)$. After developing the relevant $\mathrm{SU}(m)$ algebra, we compare two natural evolutions: the unrestricted negative gradient flow of the intrinsic-torsion energy and a Ricci-harmonic flow. We prove short-time existence and uniqueness for the Ricci-harmonic $\mathrm{H}$-flow, with arbitrary lower-order torsion-quadratic terms, for every closed subgroup $\mathrm{H}\subset\mathrm{SO}(n)$. For groups for which the projection to $\mathfrak{h}^\perp$ defines a $4$-form, including $\{1\}$, $\mathrm{SU}(2)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$, we express the negative gradient flow in Ricci-harmonic form up to explicit lower-order torsion terms and prove short-time existence and uniqueness by a modified DeTurck argument. We treat the genuinely different $\mathrm{SU}(m)$ case by a separate principal-symbol computation, proving short-time existence and uniqueness for the unrestricted negative gradient flow of $\mathrm{SU}(m)$-structures. The same computation identifies the natural negative gradient flow of $\mathrm{U}(m)$-structures as a borderline case, which cannot be made strictly parabolic by first-order diffeomorphism gauges. For the modified Ricci-harmonic flow, we derive heat-type evolution equations for the intrinsic torsion, a doubling-time estimate and Shi-type derivative estimates for $(|\mathrm{Rm}|^2+|\nabla T|^2+|T|^4)^{1/2}$, and a finite-time continuation criterion. In dimension six, we translate the formalism into the standard torsion forms of an $\mathrm{SU}(3)$-structure and describe, to highest order, the corresponding family of second-order quasilinear $\mathrm{SU}(3)$-flows.

math.DG↗

arXiv2026

The holonomy of the Obata connection on Joyce hypercomplex manifolds

We study the holonomy of the Obata connection on Joyce hypercomplex manifolds. For all such group manifolds except $\mathrm{SU}(2n+1)$, we show that the holonomy group is strictly contained in the quaternionic general linear group. The case of $\mathrm{SU}(2n+1)$ is more subtle: for every $n>1$, we show that there exist infinitely many Joyce hypercomplex structures with Obata holonomy strictly contained in $\mathrm{GL}(n(n+1),\mathbb{H})$. On the other hand, Soldatenkov showed that $\mathrm{SU}(3)$ has Obata holonomy equal to $\mathrm{GL}(2,\mathbb{H})$ \cite{Sol}, and we present here a new example on $\mathrm{SU}(5)$ with holonomy equal to $\mathrm{GL}(6,\mathbb{H})$. Finally, we investigate Joyce hypercomplex manifolds whose restricted holonomy lie in $\mathrm{SL}(n, \mathbb{H})$, yielding new compact examples of twisted Calabi-Yau manifolds.

math.DG↗

arXiv2026

Isometric solutions to the heterotic $\mathrm{G}_2$-system

In this note, we construct new solutions to the heterotic $\mathrm{G}_2$-system with non-abelian gauge group, both compact and non-compact, on certain $2$-step nilmanifolds and $3$-Sasakian manifolds. Our approach is based on an ansatz that allows us to vary both the $\mathrm{G}_2$-structure and the gauge data while keeping the underlying metric and orientation fixed. This leads, in particular, to distinct isometric solutions on the same manifold but with different gauge groups, and in some cases the resulting connection coincides with the characteristic connection of the $\mathrm{G}_2$-structure. We also investigate an $S^1$-invariant construction that yields further isometric solutions and with varying cosmological constant. Our results recover and extend several known examples solving the heterotic $\mathrm{G}_2$-system within a unified framework.

math.DG↗

arXiv2025

Flows of SU(2)-structures

This paper initiates a classification programme of flows of $\mathrm{SU}(2)$-structures on $4$-manifolds which have short-time existence and uniqueness. Our approach adapts a representation-theoretic method originally due to Bryant in the context of $\mathrm{G}_2$ geometry. We show how this strategy can also be used to deduce the number of geometric flows of a given $H$-structure; we illustrate this in the $\mathrm{G}_2$, $\mathrm{Spin}(7)$ and $\mathrm{SU}(3)$ cases. Our investigation also leads us to derive explicit expressions for the Ricci and self-dual Weyl curvature in terms of the intrinsic torsion of the underlying $\mathrm{SU}(2)$-structure. We compute the first variation formulae of all the quadratic functionals in the torsion; these provide natural building blocks for $\mathrm{SU}(2)$ gradient flows. In particular, our results demonstrate that both the negative gradient flow of the Dirichlet energy of the intrinsic torsion and the Ricci harmonic flow are parabolic after a modified DeTurck's trick.

math.DG↗

arXiv2025

Some remarks on strong $\mathrm{G}_2$-structures with torsion

A $\mathrm{G}_2$-structure on a $7$-manifold $M$ is called a $\mathrm{G}_2T$-structure if $M$ admits a $\mathrm{G}_2$-connection $\nabla^T$ with totally skew-symmetric torsion $T_φ$. If furthermore, $T_φ$ is closed then it is called a strong $\mathrm{G}_2T$-structure. In this paper we investigate the geometry of (strong) $\mathrm{G}_2T$-manifolds in relation to its curvature, $S^1$ action and almost Hermitian structures. In particular, we study the Ricci flatness condition of $\nabla^T$ and give an equivalent characterisation in terms of geometric properties of the $\mathrm{G}_2$ Lee form. Analogous results are also obtained for almost Hermitian $6$-manifolds with skew-symmetric Nijenhuis tensor. Moreover, by considering the $S^1$ reduction by the dual of the $\mathrm{G}_2$ Lee form, we show that Ricci-flat strong $\mathrm{G}_2T$-structures correspond to solutions of the $\mathrm{SU}(3)$ heterotic system on certain almost Hermitian half-flat $6$-manifolds. Many explicit examples are described and in particular, we construct the first examples of strong $\mathrm{G}_2T$-structures with $\nabla^T$ not Ricci flat. Lastly, we classify $\mathrm{G}_2$-flows inducing gauge fixed solutions to the generalised Ricci flow akin to the pluriclosed flow in complex geometry. The approach is this paper is based on the representation theoretic methods due to Bryant.

math.DG↗

arXiv2024

Examples of deformed Spin(7)-instantons/Donaldson-Thomas connections

We construct examples of deformed Hermitian Yang-Mills connections and deformed Spin(7)-instantons (also called Spin(7) deformed Donaldson-Thomas connections) on the cotangent bundle of $\mathbb{C}\mathbb{P}^2$ endowed with the Calabi hyperKähler structure. Deformed Spin(7)-instantons on cones over 3-Sasakian 7-manifolds are also constructed. We show that these can be used to distinguish between isometric structures and also between Sp(2) and Spin(7) holonomy cones. To the best of our knowledge, these are the first non-trivial examples of deformed Spin(7)-instantons.

math.DG↗

arXiv2023

Harmonic flow of quaternion-Kähler structures

We formulate the gradient Dirichlet flow of $Sp(2)Sp(1)$-structures on $8$-manifolds, as the first systematic study of a geometric quaternion-Kähler (QK) flow. Its critical condition of \emph{harmonicity} is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the $G_2$ case. We also draw several comparisons with $Spin(7)$-structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via $ε$-regularity. We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold $SU(3)$. We also study compactness and the formation of singularities.

math.DG↗

arXiv2022

Deformed $G_2$-instantons on $\mathbb{R}^4 \times S^3$

We construct explicit examples of deformed $G_2$-instantons, also called Donaldson-Thomas connections, on $\mathbb{R}^4 \times S^3$ endowed with the torsion free $G_2$-structure found by Brandhuber et al. and on $\mathbb{R}^+\times S^3 \times S^3$ endowed with the Bryant-Salamon conical $G_2$-structure. These are the first such non-trivial examples on a $G_2$ manifold. As a by-product of our investigation we also find an associative foliation of $\mathbb{R}^4\times S^3$ by $\mathbb{R}^2 \times S^1$.

math.DG↗

arXiv2022

Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds

We consider a dimensional reduction of the (deformed) Hermitian Yang-Mills condition on $S^1$-invariant Kähler Einstein $6$-manifolds. This allows us to reformulate the (deformed) Hermitian Yang-Mills equations in terms of data on the quotient Kähler $4$-manifold. In particular, we apply this construction to the canonical bundle of $\mathbb{C}\mathbb{P}^2$ endowed with the Calabi ansatz metric to find explicit abelian $SU(3)$ instantons and we show that these are determined by the spectrum of $\mathbb{C}\mathbb{P}^2$. We also find $1$-parameter families of explicit deformed Hermitian Yang-Mills connections. As a by-product of our investigation we find a coordinate expression for its holomorphic volume form which leads us to construct a special Lagrangian foliation of $\mathcal{O}_{\mathbb{C}\mathbb{P}^2}(-3)$.

math.DG↗

arXiv2022

Einstein metrics on bundles over hyperKähler manifolds

We construct explicit examples of quaternion-Kähler and hypercomplex structures on bundles over hyperKähler manifolds. We study the infinitesimal symmetries of these examples and the associated Galicki-Lawson quaternion-Kähler moment map. By performing the QK reduction we produce several explicit QK metrics. Moreover we are led to a new proof of a hyperKähler/quaternion-Kähler type correspondence. We also give examples of other Einstein metrics and balanced Hermitian structures on these bundles.

math.DG↗

arXiv2021

$S^1$-invariant Laplacian flow

The Laplacian flow is a geometric flow introduced by Bryant as a way for finding torsion free $G_2$-structures. If the flow is $S^1$-invariant then it descends to a flow of $SU(3)$-structures on a $6$-manifold. In this article we derive expressions for these evolution equations. In our search for examples we discover the first inhomogeneous shrinking solitons, which are also gradient. We also show that any compact non-torsion free soliton admits no infinitesimal symmetry.

math.DG↗

arXiv2021

Spin(7) metrics from Kähler Geometry

We investigate the $\mathbb{T}^2$-quotient of a torsion free $Spin(7)$-structure on an $8$-manifold under the assumption that the quotient $6$-manifold is Kähler. We show that there exists either a Hamiltonian $S^1$ or $\mathbb{T}^2$ action on the quotient preserving the complex structure. Performing a Kähler reduction in each case reduces the problem of finding $Spin(7)$ metrics to studying a system of PDEs on either a $4$- or $2$-manifold with trivial canonical bundle, which in the compact case corresponds to either $\mathbb{T}^4$, a K3 surface or an elliptic curve. By reversing this construction we give infinitely many new explicit examples of $Spin(7)$ holonomy metrics. In the simplest case, our result can be viewed as an extension of the Gibbons-Hawking ansatz.

math.DG↗

arXiv2020

$S^1$-quotient of $Spin(7)$-structures

If a $Spin(7)$ manifold $N^8$ admits a free $S^1$ action preserving the fundamental $4$-form then the quotient space $M^7$ is naturally endowed with a $G_2$-structure. We derive equations relating the intrinsic torsion of the $Spin(7)$-structure to that of the $G_2$-structure together with the additional data of a Higgs field and the curvature of the $S^1$-bundle; this can be interpreted as a Gibbons-Hawking-type ansatz for $Spin(7)$-structures. We focus on the three $Spin(7)$ torsion classes: torsion-free, locally conformally parallel and balanced. In particular we show that if $N$ is a $Spin(7)$ manifold then $M$ cannot have holonomy contained in $G_2$ unless $N$ is in fact a Calabi-Yau $4$-fold and $M$ is the product of a Calabi-Yau $3$-fold and an interval. We also derive a new formula for the Ricci curvature of $Spin(7)$-structures in terms of the torsion forms. We then describe this $S^1$-quotient construction in detail for the Bryant-Salamon $Spin(7)$ metric on the spinor bundle of $S^4$ and for the flat metric on $\mathbb{R}^8$.

math.DG↗