In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a
smooth map between
smooth manifolds; then the differential of ''φ,
,'' at a point ''x'' is, in some sense, the best
linear approximation of ''φ'' near ''x''. It can be viewed as a generalization of the
total derivative of ordinary calculus. Explicitly, the differential is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
from the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
of ''M'' at ''x'' to the tangent space of ''N'' at ''φ''(''x''),
. Hence it can be used to ''push'' tangent vectors on ''M'' ''forward'' to tangent vectors on ''N''. The differential of a map ''φ'' is also called, by various authors, the derivative or total derivative of ''φ''.
Motivation
Let
be a
smooth map from an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of
to an open subset
of
. For any point
in
, the
Jacobian
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
* Jacobian matrix and determinant
* Jacobian elliptic functions
* Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähle ...
of
at
(with respect to the standard coordinates) is the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
representation of the
total derivative of
at
, which is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
:
between their tangent spaces. Note the tangent spaces
are isomorphic to
and
, respectively. The pushforward generalizes this construction to the case that
is a smooth function between ''any''
smooth manifolds and
.
The differential of a smooth map
Let
be a smooth map of smooth manifolds. Given
the differential of
at
is a linear map
:
from the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
of
at
to the tangent space of
at
The image
of a tangent vector
under
is sometimes called the pushforward of
by
The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
).
If tangent vectors are defined as equivalence classes of the curves
for which
then the differential is given by
:
Here,
is a curve in
with
and
is tangent vector to the curve
at
In other words, the pushforward of the tangent vector to the curve
at
is the tangent vector to the curve
at
Alternatively, if tangent vectors are defined as
derivations acting on smooth real-valued functions, then the differential is given by
:
for an arbitrary function
and an arbitrary derivation
at point
(a
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
is defined as a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
that satisfies the
Leibniz rule Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following:
* Product rule in differential calculus
* General Leibniz rule, a generalization of the product rule
* Leibniz integral rule
* The alternating series test, al ...
, see:
definition of tangent space via derivations). By definition, the pushforward of
is in
and therefore itself is a derivation,
.
After choosing two
charts
A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...
around
and around
is locally determined by a smooth map
between open sets of
and
, and
:
in the
Einstein summation notation, where the partial derivatives are evaluated at the point in
corresponding to
in the given chart.
Extending by linearity gives the following matrix
:
Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map
at each point. Therefore, in some chosen local coordinates, it is represented by the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of the corresponding smooth map from
to
. In general, the differential need not be invertible. However, if
is a
local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.
Formal ...
, then
is invertible, and the inverse gives the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
of
The differential is frequently expressed using a variety of other notations such as
:
It follows from the definition that the differential of a
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials ...
is the composite of the differentials (i.e.,
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
ial behaviour). This is the ''chain rule'' for smooth maps.
Also, the differential of a
local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.
Formal ...
is a
linear isomorphism
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
of tangent spaces.
The differential on the tangent bundle
The differential of a smooth map ''φ'' induces, in an obvious manner, a
bundle map (in fact a
vector bundle homomorphism
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ever ...
) from the
tangent bundle of ''M'' to the tangent bundle of ''N'', denoted by ''dφ'' or ''φ''
∗, which fits into the following
commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
:
where ''π''
''M'' and ''π''
''N'' denote the bundle projections of the tangent bundles of ''M'' and ''N'' respectively.
induces a
bundle map from ''TM'' to the
pullback bundle ''φ''
∗''TN'' over ''M'' via
:
where
and
The latter map may in turn be viewed as a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of the
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
over ''M''. The bundle map ''dφ'' is also denoted by ''Tφ'' and called the tangent map. In this way, ''T'' is a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
.
Pushforward of vector fields
Given a smooth map and a
vector field ''X'' on ''M'', it is not usually possible to identify a pushforward of ''X'' by φ with some vector field ''Y'' on ''N''. For example, if the map ''φ'' is not surjective, there is no natural way to define such a pushforward outside of the image of ''φ''. Also, if ''φ'' is not injective there may be more than one choice of pushforward at a given point. Nevertheless, one can make this difficulty precise, using the notion of a vector field along a map.
A
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of ''φ''
∗''TN'' over ''M'' is called a vector field along ''φ''. For example, if ''M'' is a submanifold of ''N'' and ''φ'' is the inclusion, then a vector field along ''φ'' is just a section of the tangent bundle of ''N'' along ''M''; in particular, a vector field on ''M'' defines such a section via the inclusion of ''TM'' inside ''TN''. This idea generalizes to arbitrary smooth maps.
Suppose that ''X'' is a vector field on ''M'', i.e., a section of ''TM''. Then,
yields, in the above sense, the pushforward ''φ''
∗''X'', which is a vector field along ''φ'', i.e., a section of ''φ''
∗''TN'' over ''M''.
Any vector field ''Y'' on ''N'' defines a
pullback section ''φ''
∗''Y'' of ''φ''
∗''TN'' with . A vector field ''X'' on ''M'' and a vector field ''Y'' on ''N'' are said to be ''φ''-related if as vector fields along ''φ''. In other words, for all ''x'' in ''M'', .
In some situations, given a ''X'' vector field on ''M'', there is a unique vector field ''Y'' on ''N'' which is ''φ''-related to ''X''. This is true in particular when ''φ'' is a
diffeomorphism
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 differentiable.
Definition
Given two ...
. In this case, the pushforward defines a vector field ''Y'' on ''N'', given by
:
A more general situation arises when ''φ'' is surjective (for example the
bundle projection
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
of a fiber bundle). Then a vector field ''X'' on ''M'' is said to be projectable if for all ''y'' in ''N'', ''dφ''
''x''(''X
x'') is independent of the choice of ''x'' in ''φ''
−1(). This is precisely the condition that guarantees that a pushforward of ''X'', as a vector field on ''N'', is well defined.
Examples
Pushforward from multiplication on Lie groups
Given a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
, we can use the multiplication map
to get left multiplication
and right multiplication
maps
. These maps can be used to construct left or right invariant vector fields on
from its tangent space at the origin
(which is its associated
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
). For example, given
we get an associated vector field
on
defined by
for every
. This can be readily computed using the curves definition of pushforward maps. If we have a curve
where
and
we get
since
is constant with respect to
. This implies we can interpret the tangent spaces
as
.
Pushforward for some Lie groups
For example, if
is the Heisenberg group given by matrices
it has Lie algebra given by the set of matrices
since we can find a path
giving any real number in one of the upper matrix entries with
(i-th row and j-th column). Then, for
we have
which is equal to the original set of matrices. This is not always the case, for example, in the group
we have its Lie algebra as the set of matrices
hence for some matrix
we have
which is not the same set of matrices.
See also
*
Pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
*
Normalizing flow
References
*
* ''See section 1.6''.
* ''See section 1.7 and 2.3''.
{{Manifolds
Generalizations of the derivative
Differential geometry
Smooth functions