Tangential-valued form

Tangential-valued forms are a generalization of differential forms , in which the set of form values is a tangent bundle of a manifold .

Content

Definition

Tangential-valued form on a variety $M$ ${\ displaystyle M}$ is called the cross section of the tensor product of the tangent and external degree of cotangent bundles to a manifold:

\omega \colon M\to \left(\wedge ^{k}T^{*}M\right)\otimes _{M}TM

{\ displaystyle \ omega \ colon M \ to \ left (\ wedge ^ {k} T ^ {*} M \ right) \ otimes _ {M} TM}

\pi \circ \omega =\imath d

{\ displaystyle \ pi \ circ \ omega = \ imath d}

Operations

Internal trimming
External differentiation

Derivative Lee

A special case of tangential-valued forms are vector fields . Lee's derivative of the tensor field $T$ ${\ displaystyle T}$ along the vector field $X$ ${\ displaystyle X}$ defined in the standard way:

L_{X}T={\frac {d}{dt}}g^{t}T

{\ displaystyle L_ {X} T = {\ frac {d} {dt}} g ^ {t} T}

Where $g^{t}$ ${\ displaystyle g ^ {t}}$ - phase stream corresponding to the vector field $X$ ${\ displaystyle X}$ . This operation is related to internal multiplication. $\imath _{X}$ ${\ displaystyle \ imath _ {X}}$ differential forms on a vector field and external differentiation by the homotopy formula :

L_{X}=\imath _{X}d+d\imath _{X}

{\ displaystyle L_ {X} = \ imath _ {X} d + d \ imath _ {X}}

i.e

L_{X}=[\imath _{X},d]

{\ displaystyle L_ {X} = [\ imath _ {X}, d]}

Where $[\cdot ,\cdot ]$ ${\ displaystyle [\ cdot, \ cdot]}$ - commutator in the graded algebra of differentiations of tangential-valued forms. For an arbitrary tangential-valued form $K$ ${\ displaystyle K}$ Lee derivative is determined by analogy:

L_{K}=[\imath _{K},d]

{\ displaystyle L_ {K} = [\ imath _ {K}, d]}

Properties

$[L_{K},d]=0$ ${\ displaystyle [L_ {K}, d] = 0}$

Froehlicher-Neuenhuis bracket

Frohlichera-Neuenhuis bracket $[\cdot ,\cdot ]$ ${\ displaystyle [\ cdot, \ cdot]}$ two tangential-valued forms $K$ ${\ displaystyle K}$ and $F$ ${\ displaystyle F}$ defined as such a single tangential-valued form $[K,F]$ ${\ displaystyle [K, F]}$ , for which

[L_{K},L_{F}]=L([K,F])

{\ displaystyle [L_ {K}, L_ {F}] = L ([K, F])}

This operation is graded anticommutative and satisfies the graded Jacobi identity . If you perceive an almost complex structure $I$ ${\ displaystyle I}$ as a tangent 1-form, its Neuenhuis tensor (a tensor interfering with the search for complex local maps) is expressed in terms of the Fröhlichera-Neuenhuis bracket as $[I,I]$ ${\ displaystyle [I, I]}$ . ^{[1] The} condition of "integrability" of a certain structure as the zeroing of some of its brackets with itself is common: for example, the condition of associativity of an algebra $A$ ${\ displaystyle A}$ can be defined as the vanishing of the Gerstenhaber bracket on the codifferentiation space of a free coalgebra generated by the underlying vector space of the algebra $A$ ${\ displaystyle A}$ planted in graduation 1 (bilinear multiplications $\mu \colon A\otimes A\to A$ ${\ displaystyle \ mu \ colon A \ otimes A \ to A}$ the essence is the same as coding graduation 1) ^[2] .

Neyenhueis-Richardson bracket

Neyenhueis-Richardson bracket (algebraic brackets) $[\cdot ,\cdot ]^{\wedge }$ ${\ displaystyle [\ cdot, \ cdot] ^ {\ wedge}}$ two tangential-valued forms $K$ ${\ displaystyle K}$ and $F$ ${\ displaystyle F}$ defined as such a single tangential-valued form $[K,F]^{\wedge }$ ${\ displaystyle [K, F] ^ {\ wedge}}$ , for which

[\imath _{K},\imath _{F}]=\imath ([K,F]^{\wedge })

{\ displaystyle [\ imath _ {K}, \ imath _ {F}] = \ imath ([K, F] ^ {\ wedge})}

This operation is graded anticommutative and satisfies the graded Jacobi identity . Explicit view for the bracket of two forms $K\in \Omega ^{k+1}(M,TM)$ ${\ displaystyle K \ in \ Omega ^ {k + 1} (M, TM)}$ , $F\in \Omega ^{f+1}(M,TM)$ ${\ displaystyle F \ in \ Omega ^ {f + 1} (M, TM)}$ :

[K,F]^{\wedge }=\imath _{k}F-(-1)^{kf}\imath _{F}K

{\ displaystyle [K, F] ^ {\ wedge} = \ imath _ {k} F - (- 1) ^ {kf} \ imath _ {F} K}

Related definitions

The form is called soldering if it lies in $T^{*}M\otimes TM$ ${\ displaystyle T ^ {*} M \ otimes TM}$ .

Notes

↑ Dozen definitions of the Nijenhuis tensor $N_{J}\in \Lambda ^{2}T^{*}M\otimes TM$ ${\ displaystyle N_ {J} \ in \ Lambda ^ {2} T ^ {*} M \ otimes TM}$ complex anatomy $J\in T^{*}M\otimes TM$ ${\ displaystyle J \ in T ^ {*} M \ otimes TM}$ .
↑ Homological methods in Non-commutative Geometry, Lecture 8. , Lemma 8.2

Literature

GA Sardanashvili Modern methods of field theory. T.1: Geometry and classical fields, - M .: URSS, 1996. - 224 p.
Ivan Kolář, Peter W. Michor, Jan Slovák Natural operations in differential geometry , - Springer-Verlag, Berlin, Heidelberg, 1993. - ISBN 3-540-56235-4 , ISBN 0-387-56235-4 .

[1] Dozen definitions of the Nijenhuis tensor $N_{J}\in \Lambda ^{2}T^{*}M\otimes TM$ ${\ displaystyle N_ {J} \ in \ Lambda ^ {2} T ^ {*} M \ otimes TM}$ complex anatomy $J\in T^{*}M\otimes TM$ ${\ displaystyle J \ in T ^ {*} M \ otimes TM}$ .

[2] Homological methods in Non-commutative Geometry, Lecture 8. , Lemma 8.2