[[Geometry MOC]]
# Differential geometry MOC

**Differential geometry** studies [[Differentiability|smooth]] geometric structures, in particular [[Differentiable manifold|differentiable manifolds]].
As such it is closely related to differential topology.
A guiding principle is the [[Linearization dogma]].

## Milnor's approach

[[John Willard Milnor]] and [[Lyle Noakes]] take an approach where only [[Real embedded manifold|real embedded manifolds]] are considered, as justified by the [[Whitney embedding theorem]].

- [[Differentiability#Arbitrary subsets of real coördinate space|Differentiability on arbitrary subsets of real coördinate space]]
- [[Tangent space#Real embedded manifold|Tangent space]]
- [[Differential pushforward#Real embedded manifold|Differential pushforward]]

## General definitions

- [[Submanifold]]
- [[Immersion and submersion]]
- [[Inverse function theorem#Corollary|Inverse function theorem]] gives a condition for local diffeomorphisms
- [[Preïmage theorem]] can be used to verify most constructions that are manifolds
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