Immersion

An immersion is a smooth map whose differential is injective at every point. A parametrized surface in ${\mathbb{R}}^3$ is immersed when its two coordinate tangent vectors are linearly independent.

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 smooth embedding is an injective immersion that is a homeomorphism onto its image with the subspace topology.

A tubular neighborhood of a smooth embedded submanifold is a neighborhood identified with a neighborhood of the zero section in its normal bundle. Radial deformation of punctured fibers gives the sphere bundle, and excision identifies the associated pair with a normal disk-bundle pair.

An immersed submanifold of $M$ is a manifold $N$ with an injective immersion $\iota :N\to M$; when $N$ is treated as a subset, its manifold topology can be finer than its subspace topology.

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 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.

Coordinates $(u,v)$ on an immersed surface are isothermal when the first fundamental form is ${\lambda }^2(d{u}^2+d{v}^2)$ for a positive function $\lambda$.

For a surface of revolution whose induced metric is $g=E(\rho )d{\rho }^2+{\rho }^{2}d{\varphi }^2$, with $\rho >0$ and $E>0$ smooth on a coordinate interval, define $\sigma (\rho )={\int }_{{\rho }_{\ast }}^{\rho }\sqrt{E(r)}dr/r$. Then ${\sigma }^{\prime }>0$ and the conformal factor ${\rho }^{-1}$ gives ${\rho }^{-2}g=d{\sigma }^2+d{\varphi }^2$. This proves the rescaled metric is locally the Euclidean metric without solving a curvature equation. Poles and places where $\rho$ is not a coordinate must be treated in other charts; the angular coordinate is also understood locally.