In mathematics, a **submersion** is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.

Let *M* and *N* be differentiable manifolds and be a differentiable map between them. The map *f* is a **submersion at a point** if its differential

is a surjective linear map.^{ [1] } In this case *p* is called a **regular point** of the map *f*, otherwise, *p* is a critical point. A point is a **regular value** of *f* if all points *p* in the preimage are regular points. A differentiable map *f* that is a submersion at each point is called a **submersion**. Equivalently, *f* is a submersion if its differential has constant rank equal to the dimension of *N*.

A word of warning: some authors use the term *critical point* to describe a point where the rank of the Jacobian matrix of *f* at *p* is not maximal.^{ [2] } Indeed, this is the more useful notion in singularity theory. If the dimension of *M* is greater than or equal to the dimension of *N* then these two notions of critical point coincide. But if the dimension of *M* is less than the dimension of *N*, all points are critical according to the definition above (the differential cannot be surjective) but the rank of the Jacobian may still be maximal (if it is equal to dim *M*). The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem.

Given a submersion between smooth manifolds of dimensions and , for each there are surjective charts of around , and of around , such that restricts to a submersion which, when expressed in coordinates as , becomes an ordinary orthogonal projection. As an application, for each the corresponding fiber of , denoted can be equipped with the structure of a smooth submanifold of whose dimension is equal to the difference of the dimensions of and .

For example, consider given by The Jacobian matrix is

This has maximal rank at every point except for . Also, the fibers

are empty for , and equal to a point when . Hence we only have a smooth submersion and the subsets are two-dimensional smooth manifolds for .

- Any projection
- Local diffeomorphisms
- Riemannian submersions
- The projection in a smooth vector bundle or a more general smooth fibration. The surjectivity of the differential is a necessary condition for the existence of a local trivialization.

One large class of examples of submersions are submersions between spheres of higher dimension, such as

whose fibers have dimension . This is because the fibers (inverse images of elements ) are smooth manifolds of dimension . Then, if we take a path

and take the pullback

we get an example of a special kind of bordism, called a framed bordism. In fact, the framed cobordism groups are intimately related to the stable homotopy groups.

Another large class of submersions are given by families of algebraic varieties whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstauss family of elliptic curves is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory, such as intersection homology and perverse sheaves. This family is given by

where is the affine line and is the affine plane. Since we are considering complex varieties, these are equivalently the spaces of the complex line and the complex plane. Note that we should actually remove the points because there are singularities (since there is a double root).

If *f*: *M* → *N* is a submersion at *p* and *f*(*p*) = *q* ∈ *N*, then there exists an open neighborhood *U* of *p* in *M*, an open neighborhood *V* of *q* in *N*, and local coordinates (*x*_{1}, …, *x*_{m}) at *p* and (*x*_{1}, …, *x*_{n}) at *q* such that *f*(*U*) = *V*, and the map *f* in these local coordinates is the standard projection

It follows that the full preimage *f*^{−1}(*q*) in *M* of a regular value *q* in *N* under a differentiable map *f*: *M* → *N* is either empty or is a differentiable manifold of dimension dim *M* − dim *N*, possibly disconnected. This is the content of the **regular value theorem** (also known as the **submersion theorem**). In particular, the conclusion holds for all *q* in *N* if the map *f* is a submersion.

Submersions are also well-defined for general topological manifolds.^{ [3] } A topological manifold submersion is a continuous surjection *f* : *M* → *N* such that for all *p* in *M*, for some continuous charts ψ at *p* and φ at *f(p)*, the map ψ^{−1} ∘ f ∘ φ is equal to the projection map from *R*^{m} to *R*^{n}, where *m* = dim(*M*) ≥ *n* = dim(*N*).

- ↑ Crampin & Pirani 1994 , p. 243. do Carmo 1994 , p. 185. Frankel 1997 , p. 181. Gallot, Hulin & Lafontaine 2004 , p. 12. Kosinski 2007 , p. 27. Lang 1999 , p. 27. Sternberg 2012 , p. 378.
- ↑ Arnold, Gusein-Zade & Varchenko 1985.
- ↑ Lang 1999 , p. 27.

In mathematics, a **diffeomorphism** is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

In vector calculus, the **gradient** of a scalar-valued differentiable function *f* of several variables is the vector field whose value at a point is the vector whose components are the partial derivatives of at . That is, for , its gradient is defined at the point in *n-*dimensional space as the vector:

In mathematics, a **Lie group** is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.

In differential geometry, a subject of mathematics, a **symplectic manifold** is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

In differential geometry, a **Riemannian manifold** or **Riemannian space**(*M*, *g*) is a real, smooth manifold *M* equipped with a positive-definite inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p*. A common convention is to take *g* to be smooth, which means that for any smooth coordinate chart (*U*, *x*) on *M*, the *n*^{2} functions

In geometry, a **normal** is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the **normal line** to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one or its length may represent the curvature of the object ; its algebraic sign may indicate sides.

In the mathematical fields of differential geometry and tensor calculus, **differential forms** are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In mathematics, and particularly topology, a **fiber bundle** is a space that is *locally* a product space, but *globally* may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, , that in small regions of *E* behaves just like a projection from corresponding regions of to . The map , called the **projection** or **submersion** of the bundle, is regarded as part of the structure of the bundle. The space is known as the **total space** of the fiber bundle, as the **base space**, and the **fiber**.

In mathematics, specifically differential calculus, the **inverse function theorem** gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its *derivative is continuous and non-zero at the point*. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.

In differential geometry, a **pseudo-Riemannian manifold**, also called a **semi-Riemannian manifold**, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.

In mathematics, a **Lie groupoid** is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations are smooth, and the source and target operations

In mathematics, a **foliation** is an equivalence relation on an *n*-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension *p*, modeled on the decomposition of the real coordinate space **R**^{n} into the cosets *x* + **R**^{p} of the standardly embedded subspace **R**^{p}. The equivalence classes are called the **leaves** of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class *C ^{r}* it is usually understood that

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 **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 differential geometry, in the category of differentiable manifolds, a **fibered manifold** is a surjective submersion

In algebraic geometry, a morphism between schemes is said to be **smooth** if

In mathematics, the **rank** of a differentiable map between differentiable manifolds at a point is the rank of the derivative of at . Recall that the derivative of at is a linear map

In differential topology, a branch of mathematics, a **stratifold** is a generalization of a differentiable manifold where certain kinds of singularities are allowed. More specifically a stratifold is stratified into differentiable manifolds of (possibly) different dimensions. Stratifolds can be used to construct new homology theories. For example, they provide a new geometric model for ordinary homology. The concept of stratifolds was invented by Matthias Kreck. The basic idea is similar to that of a topologically stratified space, but adapted to differential topology.

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*Riemannian Geometry*. ISBN 978-0-8176-3490-2. - Frankel, Theodore (1997).
*The Geometry of Physics*. Cambridge: Cambridge University Press. ISBN 0-521-38753-1. MR 1481707. - Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004).
*Riemannian Geometry*(3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-20493-0. - Kosinski, Antoni Albert (2007) [1993].
*Differential manifolds*. Mineola, New York: Dover Publications. ISBN 978-0-486-46244-8. - Lang, Serge (1999).
*Fundamentals of Differential Geometry*. Graduate Texts in Mathematics. New York: Springer. ISBN 978-0-387-98593-0. - Sternberg, Shlomo Zvi (2012).
*Curvature in Mathematics and Physics*. Mineola, New York: Dover Publications. ISBN 978-0-486-47855-5.

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