# Differential Calculus on Manifolds with a Boundary. Applications

## TL;DR

This paper presents a geometric approach to differential calculus on manifolds with boundaries and corners, emphasizing applications to quantum state spaces through explicit examples.

## Contribution

It introduces a novel geometric framework for differential calculus on manifolds with boundaries, with specific focus on quantum state applications.

## Key findings

- Provides explicit examples illustrating the geometric approach.
- Develops a new method for differential calculus on manifolds with boundary.
- Highlights applications to quantum state spaces.

## Abstract

This paper contains a set of lecture notes on manifolds with boundary and corners, with particular attention to the space of quantum states. A geometrically inspired way of dealing with these kind of manifolds is presented,and explicit examples are given in order to clearly illustrate the main ideas.