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Aritra Bhowmick

Publications and source records attributed to Aritra Bhowmick.

12 recordsLinked to original sources
arXiv2026

Stability of the Injectivity Radius and the Cut Locus of Submanifolds under Perturbations

The continuity of the injectivity radius of a compact manifold under $C^2$ perturbation of the Riemannian metric was originally proved by P. Ehrlich (Composito Math., 1974), and later the proof was simplified by T. Sakai (Math. J. Okayama Univ., 1983). Using this continuity, jointly with J. Itoh and S. Prasad (J. Math. Anal. Appl., 2025), we proved the Hausdorff stability of the cut locus of a point, when both the point and the metric are perturbed. In the present article, we extend both these results to submanifolds. We first show that the injectivity radius of a submanifold depends continuously on the metric. Then, we obtain the Hausdorff stability of the cut locus of the submanifold, under $C^2$ perturbation of the metric. In fact, we allow the submanifold to be perturbed in the Whitney $C^2$ sense as well.

math.DG↗
arXiv2025

Hausdorff Stability of the Cut Locus Under $C^2$-Perturbations of the Metric

In this article, we prove the stability with respect to the Hausdorff metric $d_H$ of the cut locus $\mathrm{Cut}(p, \mathfrak{g})$ of a point $p$ in a compact Riemannian manifold $(M, \mathfrak{g})$ under $C^2$ perturbation of the metric. Specifically, given a sequence of metrics $\mathfrak{g}_i$ on $M$, converging to $\mathfrak{g}$ in the $C^2$ topology, and a sequence of points $p_i$ in $M$, converging to $p$, we show that $\lim_i d_{H}\left( \mathrm{Cut}(p_i, \mathfrak{g}_i), \mathrm{Cut}(p, \mathfrak{g}) \right) = 0$. Along the way, we also prove the continuous dependence of the cut time map on the metric.

math.DG↗
arXiv2025

On the generalized brace product: relation to $H$-splitting of loop space fibrations \& the $J$-homomorphism

Given a fibration $F \hookrightarrow E \rightarrow B$ with a homotopy section $s: B \rightarrow E$, James introduced a binary product $\left\{, \right\}_s: π_i B \times π_j F \rightarrow π_{i+j-1} F$, called the brace product, which was later generalized by Yoon. We show that the vanishing of this generalized brace product is the precise obstruction to the $H$-splitting of the loop space fibration, i.e., $ΩE \simeq ΩB \times ΩF$ as $H$-spaces. Using rational homotopy theory, we show that for rational spaces, the vanishing of the generalized brace product coincides with the vanishing of the classical James brace product, enabling us to perform the relevant computations. In addition, the notion of $J$-homomorphism is generalized and connected to the generalized brace product. Among the applications, we characterize the homotopy types of certain fibrations, including sphere bundles over spheres.

math.AT↗
arXiv2024

Distance from a Finsler Submanifold to its Cut Locus and the Existence of a Tubular Neighborhood

In this article we prove that for a closed, not necessarily compact, submanifold $N$ of a possibly non-complete Finsler manifold $(M, F)$, the cut time map is always positive. As a consequence, we prove the existence of a tubular neighborhood of such a submanifold. When $N$ is compact, it then follows that there exists an $ε> 0$ such that the distance between $N$ and its cut locus $\mathrm{Cu}(N)$ is at least $ε$. This was originally proved by B. Alves and M. A. Javaloyes (Proc. Amer. Math. Soc. 2019). We have given an alternative, rather geometric proof of the same, which is novel even in the Riemannian setup. We also obtain easier proofs of some results from N. Innami et al. (Trans. Amer. Math. Soc., 2019), under weaker hypothesis.

math.DG↗
arXiv2024

On the Focal Locus of Submanifolds of a Finsler Manifold

In this article, we investigate the focal locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. The main goal is to show that the associated normal exponential map is \emph{regular} in the sense of F.W. Warner (Am. J. of Math., 87, 1965). As a consequence, we show that the normal exponential is non-injective near any tangent focal point. Extending the ideas of Warner, we study the connected components of the regular focal locus. This allows us to identify an open and dense subset, on which the focal time maps are smooth, provided they are finite. We explicitly compute the derivative at a point of differentiability. As an application of the local form of the normal exponential map, following R.L. Bishop's work (Proc. Amer. Math. Soc., 65, 1977), we express the tangent cut locus as the closure of a certain set of points, called the separating tangent cut points. This strengthens the results from the present authors' previous work (J. Geom. Anal., 34, 2024).

math.DG↗
arXiv2024

On the Cut Locus of Submanifolds of a Finsler Manifold

In this article, we investigate the cut locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. We explore the deformation and characterization of the cut locus, extending the results of Basu and the second author (\emph{Algebraic and Geometric Topology}, 2023). Given a submanifold $N$, we consider an $N$-geodesic loop as an $N$-geodesic starting and ending in $N$, possibly at different points. This class of geodesics were studied by Omori (\emph{Journal of Differential Geometry}, 1968). We obtain a generalization of Klingenberg's lemma for closed geodesics (\emph{Annals of Mathematics}, 1959) for $N$-geodesic loops in the reversible Finsler setting.

math.DG↗
arXiv2023

DGCLUSTER: A Neural Framework for Attributed Graph Clustering via Modularity Maximization

Graph clustering is a fundamental and challenging task in the field of graph mining where the objective is to group the nodes into clusters taking into consideration the topology of the graph. It has several applications in diverse domains spanning social network analysis, recommender systems, computer vision, and bioinformatics. In this work, we propose a novel method, DGCluster, which primarily optimizes the modularity objective using graph neural networks and scales linearly with the graph size. Our method does not require the number of clusters to be specified as a part of the input and can also leverage the availability of auxiliary node level information. We extensively test DGCluster on several real-world datasets of varying sizes, across multiple popular cluster quality metrics. Our approach consistently outperforms the state-of-the-art methods, demonstrating significant performance gains in almost all settings.

cs.LG↗
arXiv2023

Existence of Horizontal Immersions in Fat Distributions

Contact structures, as well as their holomorphic and quaternionic counterparts are the primary examples of strongly bracket generating (or fat) distributions. In this article we associate a numerical invariant to corank $2$ fat distribution on manifolds, referred to as \emph{degree} of the distribution. The real distribution underlying a holomorphic contact structure is of degree $2$. Using Gromov's sheaf theoretic and analytic techniques of $h$-principle, we prove the existence of horizontal immersions of an arbitrary manifold into degree $2$ fat distributions and the quaternionic contact structures. We also study immersions of a contact manifold inducing the given contact structure.

math.DG↗
arXiv2021

On Horizontal Immersions of Discs in Fat Distributions of Type $(4,6)$

In this article we discuss horizontal immersions of discs in certain corank-$2$ fat distributions on $6$-dimensional manifolds. The underlying real distribution of a holomorphic contact distribution on a complex $3$ manifold belongs to this class. The main result presented here says that the associated nonlinear PDE is locally invertible. Using this we prove the existence of germs of embedded horizontal discs.

math.DG↗
arXiv2021

LipBaB: Computing exact Lipschitz constant of ReLU networks

The Lipschitz constant of neural networks plays an important role in several contexts of deep learning ranging from robustness certification and regularization to stability analysis of systems with neural network controllers. Obtaining tight bounds of the Lipschitz constant is therefore important. We introduce LipBaB, a branch and bound framework to compute certified bounds of the local Lipschitz constant of deep neural networks with ReLU activation functions up to any desired precision. We achieve this by bounding the norm of the Jacobians, corresponding to different activation patterns of the network caused within the input domain. Our algorithm can provide provably exact computation of the Lipschitz constant for any p-norm.

cs.LG↗
arXiv2018

Stability of certain Engel-like Distributions

In this article we introduce a higher dimensional analogue of Engel structure, motivated by the Cartan prolongation of contact manifolds. We study the stability of such structure, generalizing the Gray-type stability for Engel manifolds.

math.DG↗