In mathematics, an **immersion** is a differentiable function between differentiable manifolds whose derivative is everywhere injective.^{ [1] } Explicitly, *f* : *M* → *N* is an immersion if

- Regular homotopy
- Classification
- Existence
- Codimension 0
- Multiple points
- Examples and properties
- Immersed plane curves
- Immersed surfaces in 3-space
- Generalizations
- See also
- Notes
- References
- External links

is an injective function at every point *p* of *M* (where *T _{p}X* denotes the tangent space of a manifold

The function *f* itself need not be injective, only its derivative must be.

A related concept is that of an embedding. A smooth embedding is an injective immersion *f* : *M* → *N* that is also a topological embedding, so that *M* is diffeomorphic to its image in *N*. An immersion is precisely a local embedding – i.e., for any point *x* ∈ *M* there is a neighbourhood, *U* ⊆ *M*, of *x* such that *f* : *U* → *N* is an embedding, and conversely a local embedding is an immersion.^{ [3] } For infinite dimensional manifolds, this is sometimes taken to be the definition of an immersion.^{ [4] }

If *M* is compact, an injective immersion is an embedding, but if *M* is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.

A regular homotopy between two immersions *f* and *g* from a manifold *M* to a manifold *N* is defined to be a differentiable function *H* : *M* × [0,1] → *N* such that for all *t* in [0, 1] the function *H _{t}* :

Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for 2*m* < *n* + 1 every map *f* : *M ^{m}* →

Stephen Smale expressed the regular homotopy classes of immersions *f* : *M ^{m}* →

Morris Hirsch generalized Smale's expression to a homotopy theory description of the regular homotopy classes of immersions of any *m*-dimensional manifold *M ^{m}* in any

The Hirsch-Smale classification of immersions was generalized by Mikhail Gromov.

The primary obstruction to the existence of an immersion *i* : *M ^{m}* →

The stable normal bundle is the class of normal bundles plus trivial bundles, and thus if the stable normal bundle has cohomological dimension *k*, it cannot come from an (unstable) normal bundle of dimension less than *k*. Thus, the cohomology dimension of the stable normal bundle, as detected by its highest non-vanishing characteristic class, is an obstruction to immersions.

Since characteristic classes multiply under direct sum of vector bundles, this obstruction can be stated intrinsically in terms of the space *M* and its tangent bundle and cohomology algebra. This obstruction was stated (in terms of the tangent bundle, not stable normal bundle) by Whitney.

For example, the Möbius strip has non-trivial tangent bundle, so it cannot immerse in codimension 0 (in **R**^{2}), though it embeds in codimension 1 (in **R**^{3}).

William S.Massey ( 1960 ) showed that these characteristic classes (the Stiefel–Whitney classes of the stable normal bundle) vanish above degree *n* − *α*(*n*), where *α*(*n*) is the number of "1" digits when *n* is written in binary; this bound is sharp, as realized by real projective space. This gave evidence to the *Immersion Conjecture*, namely that every *n*-manifold could be immersed in codimension *n* − *α*(*n*), i.e., in **R**^{2n−α(n)}. This conjecture was proven by RalphCohen ( 1985 ).

Codimension 0 immersions are equivalently *relative* dimension 0 * submersions *, and are better thought of as submersions. A codimension 0 immersion of a closed manifold is precisely a covering map, i.e., a fiber bundle with 0-dimensional (discrete) fiber. By Ehresmann's theorem and Phillips' theorem on submersions, a proper submersion of manifolds is a fiber bundle, hence codimension/relative dimension 0 immersions/submersions behave like submersions.

Further, codimension 0 immersions do not behave like other immersions, which are largely determined by the stable normal bundle: in codimension 0 one has issues of fundamental class and cover spaces. For instance, there is no codimension 0 immersion **S**^{1} → **R**^{1}, despite the circle being parallelizable, which can be proven because the line has no fundamental class, so one does not get the required map on top cohomology. Alternatively, this is by invariance of domain. Similarly, although **S**^{3} and the 3-torus **T**^{3} are both parallelizable, there is no immersion **T**^{3} → **S**^{3} – any such cover would have to be ramified at some points, since the sphere is simply connected.

Another way of understanding this is that a codimension *k* immersion of a manifold corresponds to a codimension 0 immersion of a *k*-dimensional vector bundle, which is an *open* manifold if the codimension is greater than 0, but to a closed manifold in codimension 0 (if the original manifold is closed).

A ** k-tuple point** (double, triple, etc.) of an immersion

The nature of the multiple points classifies immersions; for example, immersions of a circle in the plane are classified up to regular homotopy by the number of double points.

At a key point in surgery theory it is necessary to decide if an immersion *f* : **S**^{m} → *N*^{2m} of an *m*-sphere in a 2*m*-dimensional manifold is regular homotopic to an embedding, in which case it can be killed by surgery. Wall associated to *f* an invariant *μ*(*f*) in a quotient of the fundamental group ring **Z**[π_{1}(*N*)] which counts the double points of *f* in the universal cover of *N*. For *m* > 2, *f* is regular homotopic to an embedding if and only if *μ*(*f*) = 0 by the Whitney trick.

One can study embeddings as "immersions without multiple points", since immersions are easier to classify. Thus, one can start from immersions and try to eliminate multiple points, seeing if one can do this without introducing other singularities – studying "multiple disjunctions". This was first done by André Haefliger, and this approach is fruitful in codimension 3 or more – from the point of view of surgery theory, this is "high (co)dimension", unlike codimension 2 which is the knotting dimension, as in knot theory. It is studied categorically via the "calculus of functors" by Thomas Goodwillie, John Klein, and Michael S. Weiss.

- A mathematical rose with
*k*petals is an immersion of the circle in the plane with a single*k*-tuple point;*k*can be any odd number, but if even must be a multiple of 4, so the figure 8 is not a rose. - The Klein bottle, and all other non-orientable closed surfaces, can be immersed in 3-space but not embedded.
- By the Whitney–Graustein theorem, the regular homotopy classes of immersions of the circle in the plane are classified by the winding number, which is also the number of double points counted algebraically (i.e. with signs).
- The sphere can be turned inside out: the standard embedding
*f*_{0}:**S**^{2}→**R**^{3}is related to*f*_{1}= −*f*_{0}:**S**^{2}→**R**^{3}by a regular homotopy of immersions*f*:_{t}**S**^{2}→**R**^{3}. - Boy's surface is an immersion of the real projective plane in 3-space; thus also a 2-to-1 immersion of the sphere.
- The Morin surface is an immersion of the sphere; both it and Boy's surface arise as midway models in sphere eversion.

- The Morin surface

Immersed plane curves have a well-defined turning number, which can be defined as the total curvature divided by 2π. This is invariant under regular homotopy, by the Whitney–Graustein theorem – topologically, it is the degree of the Gauss map, or equivalently the winding number of the unit tangent (which does not vanish) about the origin. Further, this is a complete set of invariants – any two plane curves with the same turning number are regular homotopic.

Every immersed plane curve lifts to an embedded space curve via separating the intersection points, which is not true in higher dimensions. With added data (which strand is on top), immersed plane curves yield knot diagrams, which are of central interest in knot theory. While immersed plane curves, up to regular homotopy, are determined by their turning number, knots have a very rich and complex structure.

The study of immersed surfaces in 3-space is closely connected with the study of knotted (embedded) surfaces in 4-space, by analogy with the theory of knot diagrams (immersed plane curves (2-space) as projections of knotted curves in 3-space): given a knotted surface in 4-space, one can project it to an immersed surface in 3-space, and conversely, given an immersed surface in 3-space, one may ask if it lifts to 4-space – is it the projection of a knotted surface in 4-space? This allows one to relate questions about these objects.

A basic result, in contrast to the case of plane curves, is that not every immersed surface lifts to a knotted surface.^{ [5] } In some cases the obstruction is 2-torsion, such as in * Koschorke's example *,^{ [6] } which is an immersed surface (formed from 3 Möbius bands, with a triple point) that does not lift to a knotted surface, but it has a double cover that does lift. A detailed analysis is given in Carter & Saito (1998a), while a more recent survey is given in Carter, Kamada & Saito (2004).

A far-reaching generalization of immersion theory is the homotopy principle: one may consider the immersion condition (the rank of the derivative is always *k*) as a partial differential relation (PDR), as it can be stated in terms of the partial derivatives of the function. Then Smale–Hirsch immersion theory is the result that this reduces to homotopy theory, and the homotopy principle gives general conditions and reasons for PDRs to reduce to homotopy theory.

- ↑ This definition is given by Bishop & Crittenden 1964 , p. 185, Darling 1994 , p. 53, do Carmo 1994 , p. 11, Frankel 1997 , p. 169, Gallot, Hulin & Lafontaine 2004 , p. 12, Kobayashi & Nomizu 1963 , p. 9, Kosinski 2007 , p. 27, Szekeres 2004 , p. 429.
- ↑ This definition is given by Crampin & Pirani 1994 , p. 243, Spivak 1999 , p. 46.
- ↑ This kind of definition, based on local diffeomorphisms, is given by Bishop & Goldberg 1968 , p. 40, Lang 1999 , p. 26.
- ↑ This kind of infinite-dimensional definition is given by Lang 1999 , p. 26.
- ↑ Carter & Saito 1998; Carter, Kamada & Saito 2004 , Remark 1.23, p. 17
- ↑ Koschorke 1979

In mathematics, **differential topology** is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the *geometric* properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the topology of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology.

**Differential geometry** is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

In mathematics, **contact geometry** is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

In differential topology, **sphere eversion** is the process of turning a sphere inside out in a three-dimensional space. Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way without cutting or tearing it or creating any crease. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false.

In mathematics, the **homotopy principle** is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as occur in the immersion problem, isometric immersion problem, fluid dynamics, and other areas.

In mathematics, specifically in topology, the operation of **connected sum** is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:

In mathematics, a **submanifold** of a manifold *M* is a subset *S* which itself has the structure of a manifold, and for which the inclusion map *S* → *M* satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.

In mathematics, **geometric topology** is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In mathematics, a **Hilbert manifold** is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting. Analogously to the finite-dimensional situation, one can define a *differentiable* Hilbert manifold by considering a maximal atlas in which the transition maps are differentiable.

In mathematics, **transversality** is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.

In the mathematical field of topology, a **regular homotopy** refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions.

In mathematics, specifically geometry and topology, the **classification of manifolds** is a basic question, about which much is known, and many open questions remain.

In mathematics, the **Riemannian connection on a surface** or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.

In mathematical study of the differential geometry of curves, the **total curvature** of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:

In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed

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- Immersion at the Manifold Atlas
- Immersion of a manifold at the Encyclopedia of Mathematics

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