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.

This is the differentiable specialization of an embedding, with an injective differential in addition to its topological embedding property.

Choose finitely many coordinate charts on a compact manifold and smooth functions supported in them, positive on a cover. Extend the displayed chart products by zero. A positive first coordinate and its matching chart coordinates separate points, while their derivatives detect every nonzero tangent vector. Thus the map is an injective immersion; compactness makes it a smooth embedding. Extra zero coordinates make its codimension positive if required.

A smooth embedding of a compact manifold remains an embedding under sufficiently small C1 metric on a smooth mapping space perturbations. A positive uniform lower bound on its differential preserves immersion. Finite coordinate neighborhoods give uniform local injectivity for nearby maps, and compactness separates the remaining pairs of points. A uniform second-derivative bound gives an explicit local Taylor estimate, although it is not necessary for the C1 openness statement.

For smooth maps from a compact manifold with a fixed Riemannian metric to Euclidean space, the displayed metric measures uniform convergence of maps and their first derivatives. For a manifold embedded in Euclidean space use its induced metric. Different fixed metrics give equivalent topologies on a compact domain.

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.

A smooth embedded submanifold has a neighborhood diffeomorphic to a neighborhood of the zero section of its normal bundle, with the zero section identified with the submanifold. One may prescribe the normal-bundle splitting of the derivative along that section. Choose a Riemannian metric making the splitting orthogonal and use its normal exponential map: its derivative there is the required isomorphism, so the inverse function theorem supplies local diffeomorphisms. A sufficiently small, possibly varying, fibre radius makes these fit into a tubular neighborhood.

A compact smooth embedding has a tubular neighborhood identified with its normal bundle. Collapse the complement of its open disk neighborhood to the basepoint and identify the remaining quotient with the Thom space of that normal bundle. Pulling back its Thom class constructs a Gysin class. For a graph embedding into a product with Euclidean space, desuspending the auxiliary coordinates gives the corresponding wrong-way map.

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.