User:Cardamomo Vincenzi

Definition of smooth manifold, examples.
A manifold is a topological space locally modeled after an open subset of an euclidean space. That is a

Definition: A smooth manifold (or just manifold) is a set $$X$$ equipped with a family of 1-1 maps $$\{\phi_j:U_j(\subseteq X)\rightarrow\mathbb R^n\}$$ so that 1) $$\{U_j\}_j$$ is a family of subsets that covers $$X$$ 2) The sets $$\phi_j(U_j)$$ are open 2) Given $$j,k$$ such that $$U_{jk}=U_j\cap U_k\neq\emptyset$$, then the map $$T_{jk}=\phi_k\phi^{-1}_j:\phi_j(U_{j}\cap U_k)\rightarrow\phi_k(U_j\cap U_k)$$, called transition map, are smooth.

The single map $$\phi_j$$ is called a chart, while the collection of all charts is called an atlas. Smooth manifolds are the objects of the category $$\mbox{SmtMnf}$$ (morphisms are smooth functions between manifolds).

Example: Trivially, every euclidean space $$\mathbb R^n$$ is a smooth manifold (the atlas consists uniquely of the identity map), with $$n\in\mathbb N$$ and $$\mathbb R^0=*$$ a set with a single element.

Example: Every open subset $$U$$ of a manifold $$X$$ has an induced atlas, with charts $$\left.\phi_j\right|_{U_j\cap U}:U_j\cap U\rightarrow\mathbb R^n$$, for all $$j$$ such that $$U_j\cap U\neq\emptyset$$. In particular every open subset of an euclidean space has an induced smooth structure and open subsets of euclidean spaces form the full subcategory $$\mbox{Cart}$$ of $$\mbox{SmtMnf}$$.

Example: It's a theorem that given a smooth map $$f:U\rightarrow\mathbb R^m$$, with $$U$$ an open subset of $$\mathbb R^n$$ (for any $$n\geqslant m$$) and a regular value $$c\in\mathbb R^m$$ (that is, the differential $$df_x$$ has rank $$m$$, for all $$x\in f^{-1}(c)$$), then $$L_c=f^{-1}(c)$$ has the structure of a smooth manifold of dimension $$n-m$$.

Vector bundles
Let $$\mbox{VectBundle}$$ be the category of vector bundles. Given a smooth manifold $$M$$, let $$\mbox{VectBundle}/M$$ be the category of vector bundles over $$M$$ and vertical morphisms between bundles (a vertical morphism is a morphism of bundles $$(f,g)$$, with $$g$$ the identity of $$M$$).

The prototype of vector bundle on $$M$$ is the tangent bundle $$(TM,M,\pi)$$ (and the dual bundle, or cotangent bundle, $$(T^*M,M,\pi^*)$$). The set of global sections of the tangent bundle is $$\mathfrak X$$ (with $$M$$ implicit), while the set of global sections of the cotangent bundle is $$\Omega^1$$.

Locally free sheaves
Given a manifold $$X$$, let $$\mbox{VectBundle}/X$$ the subcategory of vector bundles over $$X$$ (that is, vector bundles $$(E,X,p)$$) and vertical morphisms $$(f,1)$$.

Definition: Given a vector bundle $$(E,X,\pi)$$, the sheaf of sections is a sheaf on $$X$$ which sends an open $$U\subseteq X$$ into the set $$\mathit\Gamma_\pi(U)$$ of smooth maps $$\sigma:U\rightarrow E$$ such that $$\pi\circ\sigma=1_U$$.

Sending a vector bundle $$(E,X,p)$$ to its sheaf of sections define a functor from $$\mbox{VectBundle}/X$$ into the category of internal abelian groups of $$\mbox{Sh}(X)$$.

Let $$\mathcal O\in\mbox{Sh}(X)$$ be the sheaf of real valued functions, called the structural sheaf of $$X$$, which is an internal local commutative ring in $$\mbox{Sh}(X)$$. It's easy to prove that

Theorem: Every sheaf of sections is a $$\mathcal O$$-module.

Therefore $$\Gamma$$ define a functor $$\mbox{VectBundle}/X\rightarrow\mathcal O\mbox{Mod}$$ from the category of vector bundles to the category of commutative $$\mathcal O$$-modules in the category of sheaves over $$X$$. Now a trivial vector bundle is a bundle in the form $$\rho:X\times R\rightarrow X$$ and the corresponding sheaf of sections, then we can write the sheaf $$\Gamma_{\rho}$$ as a sum of $$n$$ many copies of $$\mathcal O$$ since$$\Gamma_{\rho} \simeq C^\infty(\cdot,R)\simeq C^\infty(\cdot,\mathbb R^{n})\simeq\mathcal O^n$$(where $$\dim R=n$$) and the isomorphisms are all natural.

Definition: $$\mathcal F\in\mathcal O\mbox{Mod}$$ is locally free if there is a cover $$C$$ of $$X$$ and $$n\in\mathbb N$$ such that $$\mathcal F|_{U}\simeq\mathcal O^{n}|_{U}$$ as $$\mathcal O|_{U}$$''-modules, for all $$U\in C$$. A locally free sheaf with $$n=1$$ is called a line sheaf, or invertible sheaf.''

Since vector bundles are locally isomorphic to trivial vector bundles, then sheaf of sections are locally isomorphic to sheaf of sections of trivial bundles, therefore are locally free

Theorem: Every sheaf of sections (of a vector bundle) is locally free.

Internal algebra of sheaves.
^{j_1,\cdots,j_k}T_{i_{n+1},\cdots,i_{n+m}}^{j_{k+1},\cdots,j_{k+l}}$$On the other hand, $$(i,j)$$-contraction $$c^i_j:\mathbb T^n_k\mathcal F\rightarrow\mathbb T^{n-1}_{k-1}\mathcal F$$ is defined in term of local coordinates as follows $$(c^i_jT)^{i_1,\cdots,i_{n-1}}_{j_1,\cdots,j_{k-1}} =T^{i_1,\cdots,f,\cdots,i_{n-1}} _{j_1,\cdots,f,\cdots,j_{k-1}}$$(using Einstein's notation on the index $$f$$, where the $$f$$ above is in $$i$$-th position and the $$f$$ below is in $$j$$-th position). \otimes s^{\sigma_k}$$where the sum runs over the permutations in the $$k$$-symmetrical group. In particular, notice that for any permutation $$\sigma\in S^k$$ $$s^{\sigma_1}\wedge\cdots\wedge s^{\sigma_k}=(-1)^{\text{sgn}\sigma}s^1\wedge\cdots\wedge s^k$$Given a base $$s^1,\cdots,s^n$$ for $$\mathcal F$$, the family of exterior products $$s^J=s^{j_1,\cdots,j_k}:=s^{j_1}\wedge\cdots\wedge s^{j_k},\qquad J=(j_1,\cdots,j_k) $$generates (locally) the space $\bigwedge^k\mathcal F$ (though it's not a base). If we consider the tuples with $$j_1<\cdotsn$$, so the direct sum has only a finite number of non zero elements). =(-1)^{\text{sgn}\rho}\frac{1}{k!}T_{j_{\rho_1},\cdots,j_{\rho_k}} $$This map $$\mathcal A $$ is an onto morphism $\mathbb T^k\mathcal F\rightarrow\bigwedge^k\mathcal F $. In particular, consider $T\in\bigwedge^k\mathcal F $ and $S\in\bigwedge^l\mathcal F $, and define $$T\wedge S=\mathcal A(T\otimes S)=(T\otimes S)_{[j_1,\cdots,j_{k+l}]}s^{j_1}\wedge\cdots\wedge s^{j_{k+l}} $$In particular, $$T\wedge S=(-1)^{kl}S\wedge T $$, so $$\wedge $$ is an operation on $\bigwedge \mathcal F $  making it into a graded algebra. \bigoplus_k\Omega^k$$The exterior derivative is a map $$d:\Omega^{\bullet} \rightarrow\Omega^{\bullet+1}$$ such that 1) $$d=d^0$$ is the map $$\mathcal O\rightarrow\Omega^1$$ mapping $$f\in\mathcal O(U)$$ to its differential 1-form $$df\in\Omega^1(U)$$ 2) $$dd=0$$ 3) Finally,$$d(\omega\wedge\vartheta)=d\omega\wedge\vartheta+d\vartheta\wedge\omega $$In terms of local coordinates, the differential $$d$$ maps a form $$\omega=\omega^Jdq_J\in\Omega^k(U)$$ (using Einstein notation) is $$d\omega=d\omega^J\wedge dq_J$$To prove that the last formula satisfies $$dd=0$$, we require the theorem stating that partial differential operators commute with each others. Now, given a $$\mathcal O$$-vector sheaf $$\mathcal G$$, we define the following sheaf$$\mathcal A^k\mathcal G:=\Omega^k\otimes\mathcal G$$called the sheaf of $$\mathcal G$$-valued differential $$k$$-forms. Given a local base $$s^\alpha$$ for the sheaf $$\mathcal G$$, then every element $$\omega\in \Omega^k(\mathcal G)$$ can be written as a combination $$\omega=\omega^J{}_\alpha dq_J\otimes s^\alpha,\qquad\omega^J{}_\alpha\in\mathcal O_X$$and finally, we define $$\mathcal A^\bullet \mathcal G:=\Omega^\bullet\otimes\mathcal G$$, the sheaf of $$\mathcal G$$-valued differential forms. $$ has the following representation$$df\left(\frac{\partial}{\partial q^i} \right)=\frac{\partial f_j}{\partial q^i}\frac{\partial}{\partial p^j} $$The local matrix of $$df $$ is called the Jacobian matrix of $$f $$. \frac{\partial}{\partial q^i}$$
 * Sheaf of vector fields, sheaf of differential 1-forms. Let $$\Omega^1$$ be the sheaf of sections of the cotangent bundle, called sheaf of differential 1-form, with $$d:\mathcal O \rightarrow\Omega^1$$ the map that assign to an open $$U$$ the map $$\mathcal O(U)\rightarrow\Omega^1(U)$$ sending $$f$$ to the differential 1-form $$df\in\Omega^1(U)$$. The map $$d$$ satisfy the universal property that For every real differential operator $$D:\mathcal O \rightarrow\mathcal F$$, there is a unique $$\mathcal O$$-linear map $$\alpha:\Omega^1\rightarrow\mathcal F$$ such that $$D=\alpha\circ d$$. Finally, let $$\mathfrak X$$, called the sheaf of vector fields, the sheaf of sections of the tangent bundle. In particular there is a canonical pairing $$\mathfrak X\otimes\Omega^1\rightarrow\mathcal O$$given by $$X\otimes\omega\mapsto\omega(X)$$. In local coordinates $$q$$ of $$U$$, the map $$d$$ has the following local representation $$df=\frac{\partial f}{\partial q^j}dq_j$$where $$dq_j$$ is the differential of the projection $$q\mapsto q_j$$.
 * Sheaf of units. Consider a monoid $$\mathcal A\in\mbox{Sh}(X)$$, then define the sheaf $$\mathcal A^*$$ as follows$$U\mapsto \mathcal A(U)^*$$(where $$R^*$$, for a monoid $$R$$, is the group of units), called the sheaf of units. To prove that $$\mathcal A^*$$ is a sheaf, we need to prove that $$s\in\mathcal A(U)$$ is a unit if it's locally a unit, which is simple. The sheaf $$\mathcal A^*$$ is an internal group in the category $$\mbox{Sh}(X)$$.
 * Internal Hom sheaf and Tensor product sheaf. Consider first a pair of sheaves $$\mathcal G,\mathcal E\in\mathcal O\mbox{Mod}$$, let us define the following sheaf, called sheaf of $$\mathcal O$$-morphisms $$\mathcal G\rightarrow\mathcal E$$,$$\mathbf{Hom}_\mathcal{O}(\mathcal G,\mathcal E):U\mapsto\mathbf{Hom}_{\mathcal O|_U}(\mathcal G|_U,\mathcal E|_U)$$this is the internal hom functor (restricted to $$\mathcal O$$-linear morphisms of sheaves). In particular, the dual sheaf of $$\mathcal E$$ is $$\mathcal E^\wedge:=\mathbf{Hom}_\mathcal{O}(\mathcal E,\mathcal O)$$, while the sheaf of endomorphisms of $$\mathcal E$$ is $$\mbox{End}_\mathcal{O}\mathcal E=\mathbf{Hom}_\mathcal{O}(\mathcal E,\mathcal E)$$. The groups sheaf of automorphisms is $$\mbox{Aut}_\mathcal{O}\mathcal E=(\mbox{End}_\mathcal{O}\mathcal E)^*$$
 * Matrix sheaf. Let $$M_n\mathcal O:=\mathcal O^{n^2}$$ be the $$n$$-matrix sheaf (which is a ring with classic multiplication of matrices) and $$\mathcal GL_n\mathcal O=(M_n\mathcal O)^*$$ the general linear group sheaf. An isomorphism $$\mathcal E\simeq\mathcal O^n$$ corresponds to an isomorphism of monoids$$\mbox{End}_\mathcal{O}\mathcal E\simeq\mbox{End}_\mathcal{O}\mathcal O^n\simeq M_n\mathcal O$$which restrict to an isomorphism of groups (internal to $$\mbox{Sh}(X)$$)$$\mbox{Aut}_\mathcal{O}\mathcal E\simeq\mathcal GL_n\mathcal O$$where $$\mathcal GL_n\mathcal O$$ is called the general linear group sheaf. There is also a determinant morphism $$\det:M_n\mathcal O\rightarrow\mathcal O$$, which restrict to a group homomorphism $$\det:\mathcal GL_n\mathcal O\rightarrow\mathcal O^*$$.
 * Tensor product. For any $$\mathcal F,\mathcal G\in\mathcal O\mbox{Mod}$$, define $$\mathcal F\otimes_\mathcal{O}\mathcal G$$, called tensor product sheaf, as the sheafification of the presheaf$$U\mapsto\mathcal F(U)\otimes_{\mathcal O(U)}\mathcal G(U)$$The following isomorphism of $$\mathcal O$$-modules holds:$$\mathbf{Hom}_\mathcal{O}(\mathcal F\otimes_\mathcal{O}\mathcal G,\mathcal E)\simeq\mathbf{Hom}_\mathcal{O}(\mathcal F,\mathbf{Hom}_\mathcal{O}(\mathcal G,\mathcal E))$$In particular there is a canonical $$\mathcal O$$-linear morphism $$\mathcal E\otimes_\mathcal{O}\mathbf{Hom}(\mathcal E,\mathcal G)\rightarrow\mathcal G$$, called evaluation morphism, with special case $$\mathcal G=\mathcal O$$, giving the canonical morphism $$\mathcal E\otimes_\mathcal{O}\mathcal E^\wedge\rightarrow\mathcal O$$, which is the canonical pairing.
 * Tensor sheaf. Now, let $$\mathbb T^{n}\mathcal F$$ be the $$n$$-fold tensor product of $$\mathcal F$$ (with convention $$\mathbb T^0\mathcal F=\mathcal O$$) and $$\mathbb T^n_k\mathcal F=(\mathbb T^n\mathcal F)\otimes (\mathbb T^k\mathcal F^\wedge)$$ be the sheaf of $$(n,k)$$-tensors. Product of tensor is the isomorphism$$(\mathbb T^n_k\mathcal F)\otimes (\mathbb T^m_l\mathcal F)\rightarrow\mathbb T^{n+m}_{k+l}\mathcal F$$obtained by grouping together the tensor powers of $$\mathcal F$$ and $$\mathcal F^\wedge$$. Another operation is $$(i,j)$$-contraction (for appropriate $$i,j$$) given by pairing the $$i$$-th term in $$\mathbb T^n\mathcal F$$ and the $$j$$-th term of $$\mathbb T^k\mathcal F^\wedge$$, so that we have the sequence of maps$$\mathbb T^n_k\mathcal F\rightarrow(\mathbb T^{n-1}_{k-1}\mathcal F)\otimes \mathcal F\otimes\mathcal F^\wedge\rightarrow(\mathbb T^{n-1}_{k-1}\mathcal F)\otimes\mathcal O\rightarrow\mathbb T^{n-1}_{k-1}\mathcal F$$(the second map comes from the canonical morphism $$\mathcal F\otimes\mathcal F^\wedge\rightarrow\mathcal O$$). The tensor algebra sheaf is defined as the sheaf $$\mathbb T\mathcal F=\bigoplus\mathbb T^k\mathcal F$$which is an algebra (actually a $$\mathbb Z$$-graded algebra) by the operation of tensor product $$\mathbb T^n\mathcal F\otimes\mathbb T^m\mathcal F \rightarrow\mathbb T^{n+m}\mathcal F$$ (making it a commutative $$\mathcal O$$-algebra). A special case is $$\mathcal F=\mathcal X$$, the sheaf $$\mathbb T^n_k\mathcal X\simeq\mathbb T^n\mathcal X\otimes\mathbb T^k\Omega^1=:\mathbb T^n_kM$$ is called the sheaf of $$(n,k)$$-tensor fields.
 * Local frames and tensor operations . Given a local frame $$\{s^1,\cdots,s^p\}:\mathcal O|_U\rightarrow\mathcal F|_U$$ and the corresponding dual frame $$\{s_1,\cdots,s_p\}:\mathcal O|_U \rightarrow\mathcal F^\wedge|_U$$, then a local (in the open $$U$$) $$(n,k)$$-tensor $$T$$ can be written locally as a linear combination $$T=T_{i_1,\cdots,i_n}^{j_1,\cdots,j_k}s^{i_1}\otimes\cdots\otimes s^{i_n}\otimes s_{j_1}\otimes\cdots\otimes s_{j_k} $$(again, using Einstein notation), where $$T_{i_1,\cdots,i_n}^{j_1,\cdots,j_k}\in\mathcal O|_U$$. Then, in term of the local frame, the operation of tensor product is given as $$(S\otimes T)_{i_1,\cdots,i_n,i_{n+1},\cdots,i_{n+m}}^{j_1,\cdots,j_k,j_{k+1},\cdots,j_{k+l}}=S_{i_1,\cdots,i_n}
 * Exterior algebra sheaf. Consider the sheaf $$\mathbb T^k\mathcal F$$ and let $\bigwedge^k\mathcal F$ be the exterior algebra subsheaf of those covariant tensors $$T\in\mathbb T^k\mathcal F$$ such that $$T_{j_1,\cdots,j,\cdots,i,\cdots,j_k}=-T_{j_1,\cdots,i,\cdots,j,\cdots,j_k}$$In particular, given elements $$s^1,\cdots,s^k\in\mathcal F$$, we define the following element of $\bigwedge^k\mathcal F$  $$s^1\wedge\cdots\wedge s^k:=\sum_{\sigma}\frac{1}{k!}(-1)^{\text{sgn}\sigma}s^{\sigma_1}\otimes\cdots
 * Exterior products, Exterior algebra. Now, given a tensor $T\in\mathbb T^k\mathcal F $, with local representation $$T_{j_1,\cdots,j_k}s^{j_1}\otimes\cdots\otimes s^{j_k} $$, we define the following element of the exterior algebra $$\mathcal{A}(T)=T_{[j_1,\cdots,j_k]}s^{j_1,\cdots,j_k},\qquad T_{[j_1,\cdots,j_k]}
 * Pullback of sheaves. Smooth maps $$f:X\rightarrow Y$$ induce a morphism of ringed space $$(f,f^\sharp):(X,\mathcal O_X)\rightarrow(Y,\mathcal O_Y)$$, that is, $$f^\sharp $$ is a morphism of ring sheaves $$\mathcal O_Y\rightarrow f_*(\mathcal O_X)$$, where $$f_*$$ is the functor $$\text{Sh}(X)\rightarrow\text{Sh}(Y)$$, called direct image sheaf, mapping a sheaf $$\mathcal F$$ to $$f_*(\mathcal F):U\mapsto\mathcal F(f^{-1}(U))$$The functor $$f_*$$ restrict to a functor $$\mathcal O_X\mbox{Mod}\rightarrow\mathcal O_Y\mbox{Mod}$$, with a left adjoint $$f^*$$, called inverse image sheaf, which is constructed as follows: Let $$f^{-1}:\text{Sh}(Y)\rightarrow\text{Sh}(X)$$ be the left adjoint to $$f_*$$ as a functor of sheaves, then the map $$f^\sharp$$ becomes a map $$f_\flat:f^{-1}\mathcal O_Y\rightarrow\mathcal O_X$$which means that we can make $$\mathcal O_X $$ into a $$f^{-1}\mathcal O_Y $$-module, so then we define the functor $$f^*$$ as $$f^*\mathcal F=f^{-1}(\mathcal F)\otimes_{f^{-1}\mathcal O_Y}\mathcal O_X$$The functor $$f^*$$ preserves colimits and finite limits and $$f^*(\mathcal O_Y)\simeq\mathcal O_X$$, so the canonical pairing $$\mathcal G\otimes\mathbf{Hom}(\mathcal G,\mathcal E)\rightarrow\mathcal E$$is mapped to a morphism $$f^*(\mathcal G)\otimes f^*(\mathbf{Hom}(\mathcal G,\mathcal E))\rightarrow f^*(\mathcal E)$$which corresponds to a canonical morphism $$f^*(\mathbf{Hom}(\mathcal G,\mathcal E))\rightarrow\mathbf{Hom}(f^*(\mathcal G),f^*(\mathcal E))$$for locally sheaves of finite rank, the last morphism is an isomorphism. In particular, there is a canonical isomorphism $$f^*(\mathcal E)^\wedge\simeq f^*(\mathcal E^\wedge)$$.
 * Sheaf algebra of differential forms. Take the sheaf <math *tensor="" fields="" sheaf,="" differential="" forms="" sheaf.="" consider="" the="" sheaf="" $$\mathfrak X$$ of vector fields on a manifold $$X$$. Let, respectively, the sheaf of $$k$$-vector fields, of differential $$k$$-forms, and differential forms the following sheaves $$\mathfrak X^k:=\bigwedge^k\mathfrak X,\qquad\Omega^k=\bigwedge^k\Omega^1,\qquad\Omega^\bullet=
 * Differential of a map. Consider a smooth map $$f:U\rightarrow X$$, then the composition $$\mathcal O_X\xrightarrow{f^\sharp}f_*\mathcal O_U\xrightarrow{f_*(d_X)}f_*\Omega_U$$is a differential operator. The corresponding module operator $$f^\star:\Omega_X\rightarrow f_*\Omega_U$$ is the pullback of 1-forms, while taking the dual to the map $$f_\star:f^*\Omega_X\rightarrow\Omega_U$$ (using the canonical isomorphism $$f^*(\mathcal E)^\wedge\simeq f^*(\mathcal E^\wedge)$$) is a map $$df=(f_\star)^\wedge:{\mathfrak X}_U\rightarrow f^*{\mathfrak X}_X$$corresponding to the differential of $$f$$. Given a 1-form $$\omega=\omega^jdq_j$$, the pullback operation has the following local representation $$f^\star\omega=\omega^j(f)\frac{\partial f_j}{\partial p^i}dp_i$$where $$f_j=q_j(f)$$ is the $$j$$-th component of $$f$$ in local coordinates $$q$$. On the other hand, the differential $$df
 * Lie product on vector fields. Consider then two vector fields on $$U$$ (that is, elements of $$\mathfrak X(U)$$) $$v,w:\mathcal O|_U \rightarrow\mathcal O|_U$$, we define the Lie brackets $$[v,w]:\mathcal O|_U \rightarrow\mathcal O|_U$$ as $$[v,w]=v\circ w-w\circ v$$Given local coordinates $$q=(q_1,\cdots,q_n)$$, then the Lie brackets $$[v,w]$$ has the following expression in the local frame of $$\mathcal X|_U$$$$[v,w]=\left(v_j\frac{\partial w_i}{\partial q^j}-w_j\frac{\partial v_i}{\partial q^j}\right)

Differential forms, Lie derivative and internal product
\\\mathcal X\ni w\mapsto[v,w] \\\Omega^1\ni\omega\mapsto di_v\omega+i_vd\omega \end{cases}$$The map $$\mathcal L_v$$ is called Lie differential. Given two tensors $$S,T$$, we define the Lie differential on the tensor product $$S\otimes T$$ as follows$$\mathcal L_v(S\otimes T):=\mathcal L_v(S)\otimes T+S\otimes\mathcal L_v(T)$$It can be proved that $$\mathcal L_v=di_v+i_vd$$, called Cartan formula, holds for all $$k$$-forms. In the case of the Lie differential $$\mathcal L_v\omega$$ on a differential 1-form $$\omega=\omega^jdq_j$$, the form in local coordinates is given by$$\mathcal L_v\omega=\left(v_i\frac{\partial\omega^j}{\partial q^i}+\frac{\partial v_i}{\partial q^j}\omega^i\right)dq_j$$where we used the formula $$i_v(dq_i\wedge dq_j)=v_idq_j-v_jdq_i$$.
 * Internal product. Consider a vector field $$v$$ and define the map $$i_v:\Omega^{\bullet+1}\rightarrow\Omega^\bullet$$, called interior product, as the sum of the maps $$\Omega^{p+1}\rightarrow\Omega^p$$ defined as $$i_v\vartheta:=\vartheta(v\wedge -)$$The map $$i_v$$ has the property that $$i_v\vartheta=\vartheta(v)$$ (for a 1-forms $$\vartheta$$). Given local coordinates $$q$$ on the open subset $$U$$, then $$i_v\omega$$ on a $$p+1$$-form $$\omega=\omega^Jdq_J$$ as the following local representation$$i_v\omega=v_j\omega^{j,J}dq_J$$The notion of interior product is linked to that of Lie differential, which aims to capture the idea of directional derivative.-
 * Lie differential. Now, define a map $$\mathcal L_v$$, for a fixed vector field $$v$$, on the sheaf of $$(n,k)$$-tensors $$\mathbb T^{n}_kM$$ as induction on $$n,k$$ with the basic cases $$\mathcal O,\mathcal X$$ and $$\Omega^1$$ as follows$$\mathcal L_v:\begin{cases}\mathcal O\ni f\mapsto v(f)
 * Canonical 1-form (Liouville 1-form). Consider a smooth manifold $$Q$$ and for every point $$(q,p)\in T^*Q$$ the map $$\vartheta_{(q,p)}:=p\circ d_{(q,p)}\pi^* :T_{(q,p)}T^*Q\rightarrow T_qQ\rightarrow \R$$where $$\pi^*:T^*Q\rightarrow Q$$ is the canonical projection. The correspondence $$\vartheta:T^*Q\ni(q,p)\mapsto \vartheta_{(q,p)}$$ defined a 1-form on $$T^*Q$$ called tautological 1-form, while the differential $$\omega=-d\vartheta$$ is the canonical 2-form.
 * Connections. Consider a vector sheaf $$\mathcal E$$ over $$M$$, define $$\mathcal A^k\mathcal E:=\Omega^k\otimes\mathcal E$$ (in particular, $$\mathcal A^0\mathcal E=\mathcal E$$ and $$\mathcal A^k\mathcal O=\Omega^k$$). A connection is a group homomorphism $$\nabla:\mathcal A^0\mathcal E\rightarrow\mathcal A^1\mathcal E$$ such that $$\nabla(fs)=df\otimes s+f\nabla s,\qquad f\in\mathcal O,s\in\mathcal E$$In particular, assume $$\mathcal E$$ is globally generated by sections $$e^1,\cdots,e^n\in\mathcal E$$, then we can write $$\nabla$$ as the sum of the operator $$d_\mathcal E:\mathcal E\rightarrow\mathcal A^1\mathcal E$$ defined as $$s_ie^i\mapsto ds_i\otimes e^i$$which satisfies the Leibniz rule, together with the $$\text{End}\mathcal E$$-valued 1-form $$dx^i\otimes A_i,\qquad A_i:s_je^j\mapsto s_j\nabla_{\partial_i}e^j$$So we can write every connection $$\nabla $$ on $$\mathcal E$$ locally as a sum of a connection $$d_\mathcal E$$ (which only depeds on $$\mathcal E$$) and a 1-form $$A_\nabla\in \mathcal A^1\text{End}\mathcal E$$. If $$\nabla_1,\nabla_2$$ are two connections on $$\mathcal E$$, then $$\Delta=\nabla_2-\nabla_1:\mathcal E\rightarrow\mathcal A^1\mathcal E$$ is an $$\mathcal O$$-module homomorphism, hence $$\Delta=A_{\nabla_2}-A_{\nabla_1}\in\mathcal A^1\text{End}\mathcal E$$
 * Curvature. Given a connection $$\nabla:\mathcal A^0\mathcal E\rightarrow\mathcal A^1\mathcal E$$

Riemannian metric and Electromagnetic theory
g_{JI}=g_{j_1i_1}\cdots g_{j_ki_k}$$where $$g_{ij}$$ is defined by the set of equations $$g_{ij}g^{jk}=\delta^k_i$$, where $$\delta^k_i$$ is a Kronecker delta. It's possible to prove that every point $$q$$ has an open $$U$$ and a family $$\{e^1,\cdots,e^n\}$$ of vector fields $$\mathfrak X(U)$$ such that $$\eta^{ij}=g(e^i\otimes e^j)=e^ie_j=\begin{cases}\pm 1 & i=j\\ 0 & i\neq j \end{cases}$$The matrix $$(\eta^{ij})_{ij}$$ is a diagonal matrix with $$s$$ entries are $$=-1$$ and the integer $$s$$ is called signature. \eta_{JI}=\omega^{J}\vartheta_{J},\qquad \eta_{JI}=\eta^{JI}=\eta^{j_1i_1}\cdots\eta^{j_ki_k} $$Now, we define the following family of maps $$\varepsilon^{j_1,\cdots,j_n}_g:U\rightarrow \R$$ as $$\varepsilon^{j_1,\cdots,j_n}=\begin{cases}(-1)^{\text{sgn}\sigma} & \text{if }\sigma=\begin{pmatrix}1 & \cdots & n\\ j_1 & \cdots & j_n\end{pmatrix}\text{ is a permutation}\\ 0 & \text{otherwise}\end{cases},\qquad\varepsilon_g^{j_1,\cdots,j_n}=(-1)^s\varepsilon^{j_1,\cdots,j_n}$$these maps form the Levi-Civita tensor form of the canonical volume form $$\nu_g$$$$\nu_g=\varepsilon_g^{j_1,\cdots,j_n}e_{j_1,\cdots,j_n}$$Finally, we define the map $$*:\Omega^k \longrightarrow\Omega^{n-k}$$, called Hodge star operator, as follows $$\omega^Je_J\longmapsto (-1)^si_{\omega^\flat}\nu_g=\omega_{j_1,\cdots,j_k}\varepsilon^{j_1,\cdots,j_n} e_{j_{k+1},\cdots,j_n},\qquad \varepsilon^{j_1,\cdots,j_n}=(-1)^s\varepsilon^{j_1,\cdots,j_n} _g $$We want to calculate the map $$**:\Omega^k\longrightarrow\Omega^k$$$$**\omega=(*\omega)^{J}\eta_{JJ}\varepsilon^{JI} e_{I}=\omega^{I} \underbrace{\eta_{JJ}\cdots\eta_{II}} _{=(-1)^s}\varepsilon^{IJ}\varepsilon^{JI}e_{I}=\square$$now we use the following identity deriving from the definition$$\varepsilon^{IJ}\varepsilon^{JI}=\text{sgn}\begin{pmatrix}J & I\\ I & J\end{pmatrix}=(-1)^{k(n-k)}$$so, we conclude that $$\square=(-1)^{k(n-k)+s}\omega$$, so $$**=(-1)^{k(n-k)+s}:\Omega^k\longrightarrow\Omega^k$$Now, consider $$\vartheta,\omega\in\Omega^{k}$$, then $$\vartheta \wedge*\omega=\vartheta^{J}(*\omega)^{I}e_{JI} ={\vartheta^{J}\omega_{J}}{\varepsilon^{JI}e_{JI}} =(-1)^s\langle\vartheta,\omega\rangle\nu_g $$In particular, applying last formula to $$**=(-1)^{k(n-k)+s }$$, we get that $$\langle*\omega,\vartheta\rangle \nu_g=\langle\vartheta,*\omega\rangle\nu_g=(-1)^s\vartheta\wedge({*}{*}\omega) =(-1)^{k(n-k)}\vartheta\wedge\omega=\omega\wedge\vartheta $$ $$ induced by the pseudo-riemannian metric, that is, $$\omega^j dq_j\longmapsto\omega^\flat=\omega^ig_{ij}\frac{\partial}{\partial q^j},\qquad v_j\frac{\partial} {\partial q^j}\longmapsto v^\sharp=v_ig^{ij}dq_j $$these are called musical isomorphisms. We then define the gradient operator $$\nabla:\mathcal O\rightarrow\mathfrak X $$ and divergence $$\nabla{\cdot}:\mathfrak X\rightarrow\mathcal O $$ as $$f\longmapsto(df)^\flat, \qquad v\longmapsto*d{*}(v^\sharp) $$Finally, if the canonical frame $\frac{\partial}{\partial q^j} $ is an orthonormal frame for $$g $$, then $$\nabla $$ on $$\mathcal O $$, $$\nabla{\cdot} $$ on $$\mathfrak X $$ and Hodge-Laplace operator $$\square $$ on $$\mathcal O $$ have the following local form $$\nabla f=\frac{\partial f}{\partial q^i}g_{ij}\frac{\partial}{\partial q^j},\qquad \nabla\cdot v=(-1)^{s+1}\frac{\partial v^j}{\partial q^j}g_{jj},\qquad\square f=(-1)^{n+s+1} \frac{\partial^2f}{\partial q^{jj}}g_{jj} $$In particular, in the Minkowski space $$\mathbb R^4 $$ with coordinate system $$(t,x,y,z) $$ and pseudo-riemannian metric $$g=\text{diag}(c^2,-1,-1,-1) $$, then $$\square f=\frac{1}{c^2}\frac{\partial^2f}{\partial t^2}-{\frac{\partial^2f}{\partial x^2} -\frac{\partial^2f}{\partial y^2}-\frac{\partial^2f}{\partial z^2}}= \frac{1}{c^2}\frac{\partial^2f}{\partial t^2}-\Delta f $$ $$ we define the following coordinate system $$\{x_\mu\}_{\mu} $$ as $$x_0=ct,\qquad(x_1,x_2,x_3)=(x,y,z) $$then the metric becomes $$g=\text{diag}(1,-1,-1,-1) $$. Now we define the field strength tensor (contravariant form)$$f^{\mu\nu}=\begin{pmatrix} 0 & \mathbf e^1 & \mathbf e^2 & \mathbf e^3 \\ -\mathbf e^1 & 0 & -\mathbf b^3 & \mathbf b^2 & \\ -\mathbf e^2 & \mathbf b^3 & 0 & -\mathbf b^1\\ -\mathbf e^3 & -\mathbf b^2 & \mathbf b^1 & 0\end{pmatrix}
 * Riemannian manifold. A Pseudo-Riemmanian manifold is a smooth manifold $$Q$$ together with a symmetrical map (called pseudo-riemannian metric) $$g:TQ\otimes TQ\longrightarrow \R,\qquad g=g^{ij}dq_i\otimes dq_j$$such that it is fiber-wise non-degenerate. Since $$g$$ is non-degenerate, it induces an isomorphism of vector bundles $$TQ\longrightarrow T^*Q$$which also induces a map $$\mathfrak X^k\rightarrow\Omega^k$$ sending a vector field $$v$$ to the $$k$$-form $$v^\sharp\in\Omega^k$$ such that $$v^\sharp=v^Jdq_J,\qquad v^{J}=v_{I}g^{IJ},\qquad g^{IJ}=g^{i_1j_1}\cdots g^{i_kj_k}$$Also, the inverse image $$\Omega^k\rightarrow \mathfrak X^k$$ sends a form $$\omega=\omega^Jdq_J$$ into the vector field $$\omega^\flat$$ with local representation $$\omega^\flat=\omega_J\frac{\partial}{\partial q^J},\qquad\omega_{J}=g_{JI}\omega^{I},\qquad
 * Volume form, Hodge star operator. Given an oriented pseudo-riemannian manifold $$(Q,g)$$ with a local orthogonal frame $$\{e^j\}_j\subseteq\mathfrak X(U)$$, we define the volume form $$\nu_g$$ as $$v^1\wedge\cdots\wedge v^n\longmapsto \det(e_jv^i)_{ij}$$In local coordinates, the previous volume form, called canonical volume form, has the following representation $$\nu_g=(-1)^se_1\wedge\cdots\wedge e_n $$where the base $$e^1,\cdots,e^n $$ is ordered so that it is oriented accordingly with the orientation of $$Q $$. Now we define the following map $$\langle-,-\rangle:\Omega^k\otimes\Omega^k\rightarrow\mathcal O$$ as $$\omega\otimes\vartheta\longmapsto \omega^J\vartheta^{I}
 * Dual Hodge differential, gradient, divergence and Hodge-Laplace operator. Now, we define the following map $$d^*:\Omega^k\rightarrow\Omega^{k-1} $$, called Hodge dual of $$d $$, as$$d^*\omega=(-1)^{nk-n+s+1}{*}d{*}\omega $$which has the following property: Defining the map $$\langle\!\langle-,-\rangle\!\rangle:\Omega^k\times\Omega^k\rightarrow\R $$ as $\langle\!\langle\omega,\vartheta\rangle\!\rangle={\int}_Q\omega\wedge*\vartheta $, then the dual map $$d^* $$ satisfy the condition $$\langle\!\langle d^*\omega,\vartheta\rangle\!\rangle=\langle\!\langle \omega,d\vartheta\rangle\!\rangle $$Finally, define the Hodge-Laplace operator $$\square:=dd^*+d^*d:\Omega^k\rightarrow\Omega^k $$. Consider the maps $$\Omega^1\longleftrightarrow\mathfrak X
 * Electromagnetic field, Maxwell's equations. In the Minkowski space $$\mathbb R^4

$$the covariant form of the field strength tensor is $$f_{\mu\nu}=g_{\mu\eta}g_{\nu\xi}f^{\eta\xi}

$$we also define the relativistic current density covector $$j^\mu=(\rho,-\mathbf j)

$$where $$\rho

$$ is the charge density, while $$\mathbf j

$$ is the current density, with covariant form $$j_\mu=g_{\mu\nu}j^\nu=(\rho,\mathbf j)

$$. Maxwell's equations are the following set of four equation for the electric and magnetic fields $$\mathbf e,\mathbf b$$ $$\nabla\times\mathbf e+\frac{\partial\mathbf b}{\partial x^0}=0,\qquad\nabla\cdot\mathbf b=0,\qquad \nabla\times\mathbf b-\frac{\partial\mathbf e}{\partial x^0}=\mathbf j,\qquad \nabla\cdot\mathbf e=\rho$$Using the field strength tensor we can rewrite the these equations in the following forms$$\frac{\partial f^{\mu\nu}}{\partial x^\eta}+\frac{\partial f^{\nu\eta}}{\partial x^\mu} +\frac{\partial f^{\eta\mu}}{\partial x^\nu}=0,\qquad \frac{\partial f_{\nu\mu}}{\partial x^\nu}= j_\mu$$Now, we define the following 2-form and 1-form $$f:=f^{\mu\nu}dx_\mu\wedge dx_\nu,\qquad j:=j^\mu dx_\mu$$The first equation above is clearly $$df=0$$, while the second equation is, with some calculations, proven to be equal to $$d^*f=-j$$, so Maxwell's equations are equivalent to the following two $$df=0,\qquad d^*f=-j$$In particular, the second equation implies $$d{*}f=-*j\rightarrow d{*}j=0\rightarrow d^*j=0$$, which is the continuity equation for the current density. In local coordinates, the equation $$d^*j=0$$ is equivalent to $$\frac{\partial j_\mu}{\partial x^\mu}=\frac{\partial\rho}{\partial x^0}-\nabla\cdot\mathbf j=0$$this equations expresses that flux of current through the surface enclosing a volume $$V$$ is balanced by the variation in time of the charged within that volume, that is, $$\iint_{\partial V}\mathbf j=\iiint_V\nabla\cdot\mathbf j=\frac{\partial \rho_V}{\partial t}, \qquad \rho_V =\iiint_V\rho$$Since $$\R^4$$ is a contractible space, $$df=0$$ implies that $$f\in\Omega^2$$ is exact, that is, there is a 1-form $$A\in\Omega^1$$, called electromagnetic potential, such that $$dA=f$$. The electromagnetic potential $$A$$ is not uniquely determined, since for another electromagnetic potential $$B\in\Omega^1$$ $$d(A-B)=dA-dB=f-f=0$$so again, since $$\R^4$$ is contractible, there is a 0-form $$g\in\Omega^0=\mathcal O$$ such that $$A-B=dg$$. $$the first is $$2(\mathbf e^2-\mathbf b^2)dx^0\wedge dx^1\wedge dx^2\wedge dx^3$$, while the second is equal to $$-2(\mathbf e\cdot\mathbf b)dx^0\wedge dx^1\wedge dx^2\wedge dx^3$$. An electromagnetic field is isotropic if the two 4-forms are zero, which is equivalent to $$\mathbf e\cdot\mathbf b=0 $$ and $$\mathbf e^2=\mathbf b^2 $$, that is, the electric and magnetic field are perpendicular and with equal amplitude.
 * Gauge conditions, Lorentz invariants. A single electromagnetic potential is fixed by imposing the so-called gauge conditions. For example, once condition can be the Lorentz gauge $$d^*A=0$$The existence of an electromagnetic potential $$A$$ for $$f$$ implies that the equation $$df=0$$ is satisfied, while the second equation $$d^*f$$ is equivalent, in terms of $$A$$, to (assuming the Lorentz gauge $$d^*A=0$$) $$-j=d^*dA=(d^*{d}+dd^*)A=\square A$$this is called a wave equation. Two important Lorentz invariant 4-forms are the following $$f\wedge *f,\qquad f\wedge f$$These 4-forms are, in local coordinates, equal to $$f_{\mu\nu}f^{\mu\nu}dx^0\wedge dx^1\wedge dx^2\wedge dx^3,\qquad f_{\mu\nu}(*f^{\mu\nu})dx^0\wedge dx^1\wedge dx^2\wedge dx^3

Symplectic Linear Algebra
-\mathbf 1_n & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}$$Let $$J_n$$ be the non degenerate $$2n\times 2n$$ sub-matrix. The base is constructed by an inductive method on the dimension of the space $$\mathcal E$$. In particular, notice that if $$\omega$$ is non-degenerate, then $$\mathcal E$$ has even dimension. If $$\omega$$ is non-degenerate, then we can define the following form $$\Omega=(-1)^{\frac{n(n-1)}{2}}\omega^n,\qquad\omega^n=\underbrace{\omega\wedge\cdots\wedge\omega}_{n\text{ times}}$$called canonical volume form. \omega^n $$
 * Symplectic space. Given a finite-dimensional vector space $$\mathcal E$$ with base $$e^j$$ and dual base $$e_j$$. Consider $$\omega\in\bigwedge^2\mathcal E^\wedge,\qquad \omega=\omega^{ij}e_i\wedge e_j$$we define the map $$\flat:\mathcal E\rightarrow \mathcal E^\wedge$$, called flat homomorphism, as $$\flat e^i=\omega^{ij}e_j$$If $$\flat$$ is an isomorphism, denote by $$\omega_{ij}$$ the inverse matrix to $$\omega^{ij}$$ and $$\sharp:\mathcal E^\wedge\rightarrow \mathcal E$$, called sharp homomorphism. Every pseudo-symplectic space $$(\mathcal E,\omega)$$ admits a base $$f_j$$ such that $$\omega$$ has the following matrix representation is said base $$\begin{pmatrix} 0 & \mathbf 1_n & 0\\
 * Symplectic group. Given a map $$f:\mathcal E\rightarrow \mathcal G$$ is a symplectic map $$(\mathcal E,\omega)\rightarrow(\mathcal G,\rho)$$ if it pullbacks the form $$\rho$$ to $$\omega$$, that is, $$f^*\rho=\omega,\qquad \rho\circ[\wedge^2 f]=\omega,\qquad f^i{}_\mu f^j{}_\nu\rho^{\mu\nu}=\omega^{ij} $$where $$(f^j{}_\mu) $$ is a matrix associated to $$f $$. Also, given a symplectomorphism $$f $$, then this map preserves the canonical base form $$(-1)^{\frac{n(n-1)}{2}}f^*(\rho^n)=(-1)^{\frac{n(n-1)}{2}}(f^*\rho)^n=(-1)^{\frac{n(n-1)}{2}}
 * Compatible complex structure. Given a space $$\mathcal E $$, a complex structure is an endomorphism $$J:\mathcal E\rightarrow\mathcal E $$ such that $$J^2=-1 $$. With a complex structure, we can define a $$\C $$-linear structure as $$(a+b\boldsymbol j)v=av+b(Jv) $$(where $$\boldsymbol j $$ is the imaginary unit). A complex structure $$J $$ is compatible with a symplectic form $$\omega $$ if the following form $$ m_{J}=-\omega^{ij}u_i\wedge(J^\wedge u_j) $$is a real scalar product. Finally, the space $$\mathcal E  $$ also has a definite Hermitian form over $$\C  $$ as follows $$ h_J= m_J+\boldsymbol j\omega,\qquad  h_J=( m^{ij}_J+\boldsymbol{j}\omega^{ij})u_i\wedge u_j  $$

Symplectic Geometry
$$
 * Cotangent bundle. Given a manifold $$Q$$, then we can define a canonical 1-form on the cotangent bundle $$T^*Q$$$$\vartheta_{(q,p)}=p\circ d_{(q,p)}\pi^*:T_{(q,p)}T^*Q\rightarrow \R,\qquad\vartheta=p^jdq_j $$where $$\pi^*:T^*Q\rightarrow Q$$ is the canonical projection. Then it's easy to see that the differential form$$\omega=d\vartheta=dp^j\wedge dq_j$$is a non-degenere differential 2-form, called canonical symplectic form on $$T^*Q$$. Now, we recall that given a morphism $$f:X\rightarrow Q$$ and a 1-form $$\omega=\omega^jdq_j$$, we can define $$f^\star\omega\in\Omega_X$$ as $$f^\star\omega=\omega^j(f)\frac{\partial f_j}{\partial u^i}du_i$$then it is possible to prove that $$\vartheta$$ is the unique 1-form such that for every 1-form $$\varphi\in\Omega_X$$ (using that for every form $$\lambda$$, if $$\varphi^\star\lambda=0$$ for all 1-forms $$\varphi$$, then $$\lambda=0$$)$$\varphi^\star\vartheta=\varphi

Symmetric Algebra

 * Graded symmetric algebra. Given a $$\Z$$-graded $$\R$$-vector space $$V$$, consider the tensor algebra $$T(V)=\R\oplus V\oplus V^{\otimes 2}\oplus \cdots$$with the grading defined as follows: $$|x_1\otimes\cdots\otimes x_n|=|x_1|+\cdots+|x_n|$$(with elements of $$\R$$ having degree 0). We then quotient by imposing the graded commutative rule $$x\otimes y=(-1)^{|x||y|}y\otimes x$$. Given $$\sigma\in S_n$$ and $$x_1,\cdots,x_n\in V$$, we can define the Koszul sign $$\epsilon(\sigma)=\epsilon(\sigma;x_1,\cdots,x_n)\in\{\pm1\}$$ such that $$\epsilon(\sigma)x_{\sigma(1)}\odot\cdots\odot x_{\sigma(n)}=x_1\odot\cdots\odot x_n$$We also define $$\chi(\sigma;x_1,\cdots,x_n)=(\text{sgn}\sigma)\epsilon(\sigma;x_1,\cdots,x_n) $$ to incorporate the sign of $$\sigma $$ into the Koszul sign. Denote the quotient of $$T(V) $$ as $$S(V) $$, called the graded symmetric algebra of $$V $$. There is also an inclusion $$S(V)\hookrightarrow T(V)$$ by sending $$x_1\odot \cdots\odot x_n\mapsto\sum_{\sigma}\epsilon(\sigma;x_1,\cdots,x_n)x_{\sigma(1)}\otimes \cdots\otimes x_{\sigma(n)}$$(sum over all permutations of $$n$$ elements). In particular, take a Lie algebra $$\mathfrak g=\mathfrak g_0$$ and let $$V[1]$$ be the shifted space $$(V[k])_n=V_{n+k}$$, then the graded symmetric algebra of $$\mathfrak g^\wedge[1]$$ is the Chevalley-Eilenberg algebra $\text{CE}(\mathfrak g)=\Lambda^\bullet(\mathfrak g^\wedge)$  with differential $$\delta_\text{CE}:\Lambda^n(\mathfrak g^\wedge)\rightarrow\Lambda^{n+1}(\mathfrak g^\wedge)$$$$\delta_\text{CE}(c)(x_0,\cdots,x_n)=\sum_{i<j}(-1)^{i+j}c([x_i,x_j],x_0,\cdots,\widehat{x_i},\cdots,\widehat{x_j},\cdots,x_n)$$(where we identify $$\Lambda^n(\mathfrak g^\wedge)$$ with the space of alternating multilinear functionals on $$\mathfrak g$$).
 * Weil algebra. A $$\mathfrak g$$-dg algebra is a differential ($$\N$$-)graded algebra $$\mathcal A$$ together with a family of operators $$i_\xi,L_\xi$$, for $$\xi \in\mathfrak g$$, satisfying certain conditions. A connection on a $$\mathfrak g$$-dg algebra $$\mathcal A$$ consists of an element $$\mathcal A^1\otimes\mathfrak g$$ (where $$\mathcal A^1$$ are the $$y\in \mathcal A$$ with degree 1), which is equivalent to a morphism $$\theta:\mathfrak g^*\rightarrow \mathcal A^1$$

$$n$$-plectic Geometry

 * $$n$$-plectic geometry. A pre-$$n$$-plectic manifold is a manifold $$M$$ together with a closed $$(n+1)$$-form $$\omega$$. A pair $$(v,\lambda)$$, consisting of a vector field $$v$$ and $$(n-1)$$-form $$\lambda$$, is called Hamiltonian (or, $$v$$ is an Hamiltonian vector field) if $$d \lambda=-i_v\omega$$A vector field $$v$$ is called a local Hamiltonian vector field if $$\mathcal L_v\omega=0(\Leftrightarrow di_v\omega=0\Leftrightarrow i_v\omega\text{ closed})$$If $$v$$ is an Hamiltonian vector field, then clearly is also locally an Hamiltonian vector field. Denote by $$\mathfrak X_\text{H}$$ the presheaf of Hamiltonian vector fields and $$\mathfrak X_{\ell\text{H}}$$ the sheaf of local Hamiltonian vector fields. Clearly $$\mathfrak X_{\ell\text {H}}$$ is the kernel of $$\mathfrak X\rightarrow\Omega^{n+1},\qquad v\mapsto\mathcal L_v\omega$$The reason why $$\mathfrak X_\text{H}$$ is only a presheaf is that every local Hamiltonian vector field is locally Hamiltonian, meaning that if $$v\in\mathfrak X_{\ell\text{H}}(U)$$, then there is an open cover $$\{U_i\}_i$$ such that $$v|_{U_i}\in \mathfrak X_\text{H}(U_i)$$, which means that the inclusion $$\mathfrak X_\text{H}\hookrightarrow \mathfrak X_{\ell\text{H}}$$is also a local epimorphism. Then, contraction of $$\omega$$ induces a morphism $$-_\flat:\mathfrak X\rightarrow\Omega^n,\qquad v\mapsto i_v\omega$$If $$-_\flat$$ is injective, we call $$\omega$$ a $$n$$-plectic form.
 * Cotangent bundle.