In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Frölicher–Nijenhuis bracket is an extension of the
Lie bracket
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 identit ...
of
vector fields
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and direct ...
to
vector-valued differential form In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differe ...
s on a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
.
It is useful in the study of
connections
Connections may refer to:
* Connection (disambiguation), plural form
Television
* '' Connections: An Investigation into Organized Crime in Canada'', a documentary television series
* ''Connections'' (British TV series), a 1978 documentary tele ...
, notably the
Ehresmann connection
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does ...
, as well as in the more general study of
projections in the
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
.
It was introduced by
Alfred Frölicher
Alfred Frölicher (often misspelled Fröhlicher) was a Swiss mathematician (8 October 8 1927 – 1 July 2010). He was a full professor at the Université de Fribourg (1962–1965), and then at the Université de Genève (1966–1993). He introdu ...
and
Albert Nijenhuis
Albert Nijenhuis (November 21, 1926 – February 13, 2015) was a Dutch-American mathematician who specialized in differential geometry and the theory of deformations in algebra and geometry, and later worked in combinatorics.
His high school st ...
(1956) and is related to the work of
Schouten (1940).
It is related to but not the same as the
Nijenhuis–Richardson bracket and the
Schouten–Nijenhuis bracket.
Definition
Let Ω*(''M'') be the
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...
of
exterior algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
s of
differential form
In mathematics, 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 ...
s on a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
''M''. This is a
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...
in which forms are graded by degree:
:
A
graded derivation
In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geom ...
of degree ℓ is a mapping
:
which is linear with respect to constants and satisfies
:
Thus, in particular, the
interior product
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, or inner derivation) is a degree −1 (anti)derivation on the exterio ...
with a vector defines a graded derivation of degree ℓ = −1, whereas the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
is a graded derivation of degree ℓ = 1.
The vector space of all derivations of degree ℓ is denoted by Der
ℓΩ*(''M''). The direct sum of these spaces is a
graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a ''grading'' or ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers.
For ...
whose homogeneous components consist of all graded derivations of a given degree; it is denoted
:
This forms a
graded Lie superalgebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operati ...
under the anticommutator of derivations defined on homogeneous derivations ''D''
1 and ''D''
2 of degrees ''d''
1 and ''d''
2, respectively, by
:
Any
vector-valued differential form In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differe ...
''K'' in Ω
''k''(''M'', T''M'') with values in the
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
of ''M'' defines a graded derivation of degree ''k'' − 1, denoted by ''i''
''K'', and called the insertion operator. For ω ∈ Ω
ℓ(''M''),
:
The
Nijenhuis–Lie derivative along ''K'' ∈ Ω
k(''M'', T''M'') is defined by
:
where ''d'' is the exterior derivative and ''i''
K is the insertion operator.
The Frölicher–Nijenhuis bracket is defined to be the unique vector-valued differential form
:
such that
:
Hence,
:
If ''k'' = 0, so that ''K'' ∈ Ω
0(''M'', T''M'')
is a vector field, the usual homotopy formula for the Lie derivative is recovered
:
If ''k''=''ℓ''=1, so that ''K,L'' ∈ Ω
1(''M'', T''M''),
one has for any vector fields ''X'' and ''Y''
:
If ''k''=0 and ''ℓ''=1, so that ''K=Z''∈ Ω
0(''M'', T''M'') is a vector field and ''L'' ∈ Ω
1(''M'', T''M''), one has for any vector field ''X''
:
An explicit formula for the Frölicher–Nijenhuis bracket of
and
(for forms φ and ψ and vector fields ''X'' and ''Y'') is given by
:
Derivations of the ring of forms
Every derivation of Ω
*(''M'') can be written as
:
for unique elements ''K'' and ''L'' of Ω
*(''M'', T''M''). The Lie bracket of these derivations is given as follows.
*The derivations of the form
form the Lie superalgebra of all derivations commuting with ''d''. The bracket is given by
::
:where the bracket on the right is the Frölicher–Nijenhuis bracket. In particular the Frölicher–Nijenhuis bracket defines a
graded Lie algebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operati ...
structure on
, which extends the
Lie bracket
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 identit ...
of
vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s.
*The derivations of the form
form the Lie superalgebra of all derivations vanishing on functions Ω
0(''M''). The bracket is given by
::
:where the bracket on the right is the
Nijenhuis–Richardson bracket.
*The bracket of derivations of different types is given by
::
: for ''K'' in Ω
k(''M'', T''M''), ''L'' in Ω
l+1(''M'', T''M'').
Applications
The
Nijenhuis tensor
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...
of an
almost complex structure
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...
''J'', is half the Frölicher–Nijenhuis bracket of ''J'' with itself. An almost complex structure is a complex structure if and only if the Nijenhuis tensor is zero.
With the Frölicher–Nijenhuis bracket it is possible to define the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
and
cocurvature of a vector-valued 1-form which is a
projection
Projection or projections may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphics, and carto ...
. This generalizes the concept of the curvature of a
connection
Connection may refer to:
Mathematics
*Connection (algebraic framework)
*Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold
* Connection (affine bundle)
*Connection (composite bun ...
.
There is a common generalization of the Schouten–Nijenhuis bracket and the Frölicher–Nijenhuis bracket; for details see the article on the
Schouten–Nijenhuis bracket.
References
*.
*.
*
*.
{{DEFAULTSORT:Frolicher-Nijenhuis Bracket
Bilinear maps
Differential geometry