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Mahuya Datta

Publications and source records attributed to Mahuya Datta.

8 recordsLinked to original sources
arXiv2020

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↗
arXiv2016

Nash twist and Gaussian noise measure on isometric $C^1$ maps

Starting with a short map $f_0:I\to \mathbb R^3$ on the unit interval $I$, we construct random isometric map $f_n:I\to \mathbb R^3$ (with respect to some fixed Riemannian metrics) for each positive integer $n$, such that the difference $(f_n - f_0)$ goes to zero in the $C^0$ norm. The construction of $f_n$ uses the Nash twist. We show that the distribution of $ n^{1/2} (f_n - f_0)$ converges (weakly) to a Gaussian noise measure.

math.PR↗
arXiv2016

A Measure on the space of Lipschitz isometric maps of a compact 1-manifold into $\mathbb R^2$

Let $M$ be a compact 1-manifold. Given a continuous function $g:M\to \mathbb R_+$ we consider the following ordinary differential equation: $\|\dot{f}(t)\|=g(t)$, where $f:M\to \mathbb R^2$. We construct a probability measure on the space of almost everywhere differentiable solutions of this differential equation and study this measure. A solution of this equation can be viewed as an isometric immersion of a compact 1-manifold into $\mathbb R^2$. Nash's convergence technique in the proof of isometric $C^1$-immersion theorem plays an important role in the construction.

math.PR↗
arXiv2013

On Existence of Regular Jacobi Structures

We prove $h$-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We interpret the result on $h$ principle of contact foliations in terms of the regular Jacobi structures.

math.DG↗
arXiv2012

Immersions in a Quaternionic Grassmannian inducing a given 4-form

Let $Gr_k(\H^n)$ be the Grassmannian manifold of Quaternionic $k$-planes in $\H^n$ and let $γ^n_k\to Gr_k(\H^n)$ denote the Stiefel bundle of quaternionic $k$-frames in $\H^n$. Let $σ$ denote the first symplectic Pontrjagin form associated with the universal connection on $γ^n_k$. We show that every 4-form $ω$ on a smooth manifold $M$ can be induced from $σ$ by a smooth immersion $f:M\to Gr_k(\H^n)$ (for sufficiently large $k$ and $n$) provided there exists a continuous map $f_0:M\to Gr_k(\H^n)$ which pulls back the cohomology class of $σ$ onto that of $ω$.

math.DG↗
arXiv2012

Partial Isometries of a Sub-Riemannian Manifold

In this paper, we obtain the following generalisation of isometric $C^1$-immersion theorem of Nash and Kuiper. Let $M$ be a smooth manifold of dimension $m$ and $H$ a rank $k$ subbundle of the tangent bundle $TM$ with a Riemannian metric $g_H$. Then the pair $(H,g_H)$ defines a sub-Riemannian structure on $M$. We call a $C^1$-map $f:(M,H,g_H)\to (N,h)$ into a Riemannian manifold $(N,h)$ a {\em partial isometry} if the derivative map $df$ restricted to $H$ is isometric; in other words, $f^*h|_H=g_H$. The main result states that if $\dim N>k$ then a smooth $H$-immersion $f_0:M\to N$ satisfying $f^*h|_H<g_H$ can be homotoped to a partial isometry $f:(M,g_H)\to (N,h)$ which is $C^0$-close to $f_0$. In particular we prove that every sub-Riemannian manifold $(M,H,g_H)$ admits a partial isometry in $\R^n$ provided $n\geq m+k$.

math.DG↗
arXiv2007

Smooth maps of a foliated manifold in a symplectic manifold

The immersions of a smooth manifold $M$ in a symplectic manifold $(N,σ)$ inducing a given closed form $ω$ on $M$ satisfy the $C^0$-dense $h$-principle in the space of all continuous maps which pull back the deRham cohomology class of $σ$ onto that of $ω$. In this paper we prove a foliated version of this result due to Gromov.

math.DG↗