In this article, we prove the stability with respect to the Hausdorff metric $d_H$ of the cut locus $\mathrm{Cu}(p, \mathfrak{g})$ 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 $\mathfrak{g}$igi​ 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{Cu}(p_i, \mathfrak{g}_i), \mathrm{Cu}(p, \mathfrak{g}) \right) = 0. Along the way, we also prove the continuous dependence of the cut time map on the metric.

Given a fibration $F \hookrightarrow E \rightarrow B$ with a homotopy section $s : B \rightarrow E$, James introduced a binary product $\left{, \right}$s: \pi_i B \times \pi_j F \rightarrow \pi{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., $\Omega E \simeq \Omega B \times \Omega F$ 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.

Given a smooth bracket-generating distribution $\mathcal{D}$ of constant growth on a manifold $M$, we prove that maps from an arbitrary manifold $\Sigma$ to $M$, which are transverse to $\mathcal{D}$, satisfy the complete $h$-principle. This partially settles a question posed by M. Gromov.

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 (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 (Journal of Differential Geometry, 1968). We obtain a generalization of Klingenberg's lemma for closed geodesics (Annals of Mathematics, 1959) for $N$-geodesic loops in the reversible Finsler setting.

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 emphdegree 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.

In this article we discuss horizontal immersion of discs in certain corank $2$ fat distribution on $6$-dimensional manifolds, holomorphic contact distributions are examples of which. The main result presented here is that a certain nonlinear PDE is locally invertible, using which we will prove a local $h$-principle result for such immersions.

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 results for Engel manifolds. We also derive local normal forms defining such a distribution.