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An injective immersion from a compact smooth manifold into a Hausdorff smooth manifold is a smooth embedding. Indeed, it is a continuous bijection onto its image, and compactness makes its inverse continuous.

The map descends to an injective immersion from $\mathbb{R}\times (\mathbb{R}/\mathbb{Z})$ into the three-dimensional torus. Injectivity follows because an integer difference in $s$ with an integer difference in $\alpha s$ must be zero. Its image is dense by an irrational rotation of the circle, and is tangent to the commuting independent fields ${\partial }_x+\alpha {\partial }_z$ and ${\partial }_y$. It is a leaf of a regular foliation, but cannot be an embedded submanifold: in an embedding chart the coordinate plane would be locally closed yet contain a dense ambient subset.

A finite-dimensional commutative real algebra with a Euclidean multiplicative norm has dimension at most two. Automatic real bilinearity from a multiplicative norm supplies bilinearity if it was not part of the initial description. On its unit sphere, squaring has derivative $v\mapsto 2xv$, injective because there are no zero divisors. The only equal squares are antipodal pairs, so the map induces an injective immersion ${\mathbb{RP}}^{n-1}\to S^{n-1}$. Compactness and the inverse function theorem make it a diffeomorphism when $n\ge 2$. For $n\ge 3$, the fundamental groups $\mathbb{Z}/2$ and zero contradict this. The real and complex number algebras attain dimensions one and two.

The isothermal coordinates $(u,v)$ give the immersion $(u-u^3/3+u{v}^2,v-v^3/3+u^{2}v,u^2-v^2)$. Its coordinate functions are harmonic and its metric is $(1+u^2+v^2{)}^2(d{u}^2+d{v}^2)$, so it is a complete immersed minimal surface with self-intersections.

For irrational $\alpha$, the mapis an injective immersion with dense image. It is not a smooth embedding, since points with arbitrarily large parameter return arbitrarily close to its initial point in the subspace topology.

An isometric immersion is an immersion $f:M\to N$ between Riemannian manifolds whose derivative preserves the inner products of tangent vectors; equivalently, $f^{\ast }{g}_N=g_M$. Its derivative is injective and the target may have larger dimension. A local isometry additionally requires a local diffeomorphism, so its tangent maps are linear isomorphisms. For example, a sphere with its induced Riemannian metric is isometrically immersed in Euclidean space, although its positive intrinsic sectional curvature differs from that of the ambient space.

A Lagrangian immersion is an immersion $\iota :L^n\to (M^{2n},\omega )$ satisfying ${\iota }^{\ast }\omega =0$.

A Lie subgroup is a subgroup carrying a Lie group structure such that its inclusion is an injective immersion and a group homomorphism. Its Lie algebra includes in that of the ambient Lie group.