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

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

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

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

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

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

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

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

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.