# Existence of $B^k_{\alpha,\beta}$-Structures on $C^k$-Manifolds

## TL;DR

This paper introduces a new class of generalized $C^k$-manifolds called $B_{eta,eta}^{k}$-structures, explores their embedding properties, and studies conditions for their existence on standard manifolds.

## Contribution

It defines $B_{eta,eta}^{k}$-manifolds as a generalization of smooth manifolds with specific transition function regularity and establishes embedding theorems and existence conditions for these structures.

## Key findings

- Presented embedding theorems for structural presheaves $B$.
- Proved existence of $B_{eta,eta}^{k}$-structures under certain conditions.
- Showed the forgetful functor has adjoints in specific cases.

## Abstract

In this paper we introduce $B_{\alpha,\beta}^{k}$-manifolds as generalizations of the notion of smooth manifolds with $G$-structure or with $k$-bounded geometry. These are $C^{k}$-manifolds whose transition functions $\varphi_{ji}=\varphi_{j}\circ\varphi_{i}^{-1}$ are such that $\partial^{\mu}\varphi_{ji}\in B_{\alpha(r)}\cap C^{k-\beta(r)}$ for every $\vert\mu\vert=r$, where $B=(B_{r})_{r\in\Gamma}$ is some sequence of presheaves of Fr\'echet spaces endowed with further structures, $\Gamma\subset\mathbb{Z}_{\geq0}$ is some parameter set and $\alpha,\beta$ are functions. We present embedding theorems for the presheaf category of those structural presheaves $B$. The existence problem of $B_{\alpha,\beta}^{k}$-structures on $C^{k}$-manifolds is studied and it is proved that under certain conditions on $B$, $\alpha$ and $\beta$, the forgetful functor from $C^{k}$-manifolds to $B_{\alpha,\beta}^{k}$-manifolds has adjoints.