##  [Immersion](/immersion-0) 

 Definition

A smooth map f : M -> N between differentiable manifolds whose differential df_p is injective at every point p in M; locally the map embeds tangent spaces and places M into N without infinitesimal folding, though global self-intersections may occur.

 

 

 

 

 

 





## Principle

Principle

Injectivity of the differential is the organizing rule: an immersion preserves local dimension information (rank = dim M) and gives locally an embedding of tangent spaces; existence and behavior are constrained by dimension inequalities and smoothness class.

 

 

 

 

 





## Demonstration

Demonstration

An elementary example is the figure-eight immersion of S^1 into R^2 given by a suitable periodic smooth map; another is the inclusion of a submanifold, which is an immersion that is also an embedding when the inclusion is injective and a homeomorphism onto its image.

 

 

 

 

## Misapplication

Misapplication

Assuming every immersion is an embedding is a common mistake: immersions need not be globally injective or proper, so statements that require a homeomorphism onto the image (for example, transferring global topological invariants) can fail.

 

 

 

 

 





## Consequence

Consequence

When used correctly an immersion guarantees local manifold charts on the image and allows pulling back tensors and differential forms; it supports constructions that rely on local embedding, such as transversality arguments and certain surgery techniques.

 

 

 

 

## Reversal

Reversal

The conceptual opposite are maps with degenerate differentials (critical points) or submersions: rather than injectivity on tangents, one studies surjectivity (projection behavior) or failure of injectivity leading to singularities.

 

 

 

 

 





## Boundary

Boundary

Immersion applies in the smooth/differentiable category with a well-defined tangent functor; it excludes purely topological embeddings or continuous injections that lack a differentiable structure, and it depends on the differentiability class (C^k, C^, etc.).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Immersion sits in tension with embedding: both are local notions of 'inserting one manifold into another', but embedding adds global injectivity and topological compatibility; it also competes with the weaker notion of local homeomorphism in the topological category.

 

 

 

 

 





## Synthesis

Synthesis

An immersion is the local, differential-level insertion of one manifold into another: it secures injective tangential behavior everywhere, enabling local geometric identification of the domain inside the codomain while allowing global pathologies like self-intersection.