# Manifold

# Definition

A manifold is a topological space that locally resembles Euclidean space near each point. That is, around each point on the manifold there exists a neighborhood that is topographically similar to the open unit ball in $\mathbb{R}^n$.

## Formal Defintions

## Examples

# Why are manifolds useful

- Graph (function)s of functions are manifolds

## My confusion

I think one thing I’ve struggled with since encountering manifolds was truly understanding not only what they are, but why they are necessary and what they introduce that we couldn’t otherwise describe previously. For some reason the formulation just felt so foreign; while I could grasp its definition and solve problems with manifolds, they never really felt comfortable. This is likely due to lack of overall experience with them; after all,

“In mathematics, you never understand things; you just get used to them.”

– John von Neumann

Looking back now, I think I just managed to overlook the manifold’s slightly simpler origins. That is, when presented with the definitions of manifolds, they immediately feel obscure and confusing. Even when presented with simple examples and making connections to familiar concepts like graphs of functions, I still felt like there was this deeper, mysterious reason why they were here that I wouldn’t be able to understand. I think the key insight that resolves much of this confusion is realizing that there are regular situations where the notion of a manifold *arises naturally*, and we simply recognized that there was something new and not yet formal about the mathematical object we were looking at it. It was there that . In this context the seemingly obscure manifold definitions (which, with time, feel less so) feel far less contrived and just a necessary formalization of some natural phenomena we observe. Even if this wasn’t *really* how the notion of manifold came to be, it helps me to think that they could arise from such simple origins and simply recognizing that there was no way existing machinery to describe what was there.

Most of this little thought spawned from reading these few sentences in the Wikipedia article on manifolds:

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions.

Phrasing things this way just made it easy to ponder the necessity of manifolds and question my existing confusion on their origin.