The following summarizes short definitions and notations that are used in this article.

Manifold edit

${\displaystyle M}$ , ${\displaystyle N}$  are ${\displaystyle n}$ -dimensional smooth manifolds, where ${\displaystyle n\in \mathbb {N} }$ . That is, differentiable manifolds that can be differentiated enough times for the purposes on this page.

${\displaystyle p\in M}$ , ${\displaystyle q\in N}$  denote one point on each of the manifolds.

The boundary of a manifold ${\displaystyle M}$  is a manifold ${\displaystyle \partial M}$ , which has dimension ${\displaystyle n-1}$ . An orientation on ${\displaystyle M}$  induces an orientation on ${\displaystyle \partial M}$ .

We usually denote a submanifold by ${\displaystyle \Sigma \subset M}$ .

Tangent and cotangent bundles edit

${\displaystyle TM}$ , ${\displaystyle T^{*}M}$  denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold ${\displaystyle M}$ .

${\displaystyle T_{p}M}$ , ${\displaystyle T_{q}N}$  denote the tangent spaces of ${\displaystyle M}$ , ${\displaystyle N}$  at the points ${\displaystyle p}$ , ${\displaystyle q}$ , respectively. ${\displaystyle T_{p}^{*}M}$  denotes the cotangent space of ${\displaystyle M}$  at the point ${\displaystyle p}$ .

Sections of the tangent bundles, also known as vector fields, are typically denoted as ${\displaystyle X,Y,Z\in \Gamma (TM)}$  such that at a point ${\displaystyle p\in M}$  we have ${\displaystyle X|_{p},Y|_{p},Z|_{p}\in T_{p}M}$ . Sections of the cotangent bundle, also known as differential 1-forms (or covector fields), are typically denoted as ${\displaystyle \alpha ,\beta \in \Gamma (T^{*}M)}$  such that at a point ${\displaystyle p\in M}$  we have ${\displaystyle \alpha |_{p},\beta |_{p}\in T_{p}^{*}M}$ . An alternative notation for ${\displaystyle \Gamma (T^{*}M)}$  is ${\displaystyle \Omega ^{1}(M)}$ .

Differential k-forms edit

Differential ${\displaystyle k}$ -forms, which we refer to simply as ${\displaystyle k}$ -forms here, are differential forms defined on ${\displaystyle TM}$ . We denote the set of all ${\displaystyle k}$ -forms as ${\displaystyle \Omega ^{k}(M)}$ . For ${\displaystyle 0\leq k,\ l,\ m\leq n}$  we usually write ${\displaystyle \alpha \in \Omega ^{k}(M)}$ , ${\displaystyle \beta \in \Omega ^{l}(M)}$ , ${\displaystyle \gamma \in \Omega ^{m}(M)}$ .

${\displaystyle 0}$ -forms ${\displaystyle f\in \Omega ^{0}(M)}$  are just scalar functions ${\displaystyle C^{\infty }(M)}$  on ${\displaystyle M}$ . ${\displaystyle \mathbf {1} \in \Omega ^{0}(M)}$  denotes the constant ${\displaystyle 0}$ -form equal to ${\displaystyle 1}$  everywhere.

Omitted elements of a sequence edit

When we are given ${\displaystyle (k+1)}$  inputs ${\displaystyle X_{0},\ldots ,X_{k}}$  and a ${\displaystyle k}$ -form ${\displaystyle \alpha \in \Omega ^{k}(M)}$  we denote omission of the ${\displaystyle i}$ th entry by writing

${\displaystyle \alpha (X_{0},\ldots ,{\hat {X}}_{i},\ldots ,X_{k}):=\alpha (X_{0},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{k}).}$ 

Exterior product edit

The exterior product is also known as the wedge product. It is denoted by ${\displaystyle \wedge :\Omega ^{k}(M)\times \Omega ^{l}(M)\rightarrow \Omega ^{k+l}(M)}$ . The exterior product of a ${\displaystyle k}$ -form ${\displaystyle \alpha \in \Omega ^{k}(M)}$  and an ${\displaystyle l}$ -form ${\displaystyle \beta \in \Omega ^{l}(M)}$  produce a ${\displaystyle (k+l)}$ -form ${\displaystyle \alpha \wedge \beta \in \Omega ^{k+l}(M)}$ . It can be written using the set ${\displaystyle S(k,k+l)}$  of all permutations ${\displaystyle \sigma }$  of ${\displaystyle \{1,\ldots ,n\}}$  such that ${\displaystyle \sigma (1)<\ldots <\sigma (k),\ \sigma (k+1)<\ldots <\sigma (k+l)}$  as

${\displaystyle (\alpha \wedge \beta )(X_{1},\ldots ,X_{k+l})=\sum _{\sigma \in S(k,k+l)}{\text{sign}}(\sigma )\alpha (X_{\sigma (1)},\ldots ,X_{\sigma (k)})\otimes \beta (X_{\sigma (k+1)},\ldots ,X_{\sigma (k+l)}).}$ 

Directional derivative edit

The directional derivative of a 0-form ${\displaystyle f\in \Omega ^{0}(M)}$  along a section ${\displaystyle X\in \Gamma (TM)}$  is a 0-form denoted ${\displaystyle \partial _{X}f.}$ 

Exterior derivative edit

The exterior derivative ${\displaystyle d_{k}:\Omega ^{k}(M)\rightarrow \Omega ^{k+1}(M)}$  is defined for all ${\displaystyle 0\leq k\leq n}$ . We generally omit the subscript when it is clear from the context.

For a ${\displaystyle 0}$ -form ${\displaystyle f\in \Omega ^{0}(M)}$  we have ${\displaystyle d_{0}f\in \Omega ^{1}(M)}$  as the ${\displaystyle 1}$ -form that gives the directional derivative, i.e., for the section ${\displaystyle X\in \Gamma (TM)}$  we have ${\displaystyle (d_{0}f)(X)=\partial _{X}f}$ , the directional derivative of ${\displaystyle f}$  along ${\displaystyle X}$ .^[6]

For ${\displaystyle 0<k\leq n}$ ,^[6]

${\displaystyle (d_{k}\omega )(X_{0},\ldots ,X_{k})=\sum _{0\leq j\leq k}(-1)^{j}d_{0}(\omega (X_{0},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}))(X_{j})+\sum _{0\leq i<j\leq k}(-1)^{i+j}\omega ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}).}$ 

Lie bracket edit

The Lie bracket of sections ${\displaystyle X,Y\in \Gamma (TM)}$  is defined as the unique section ${\displaystyle [X,Y]\in \Gamma (TM)}$  that satisfies

${\displaystyle \forall f\in \Omega ^{0}(M)\Rightarrow \partial _{[X,Y]}f=\partial _{X}\partial _{Y}f-\partial _{Y}\partial _{X}f.}$ 

Tangent maps edit

If ${\displaystyle \phi :M\rightarrow N}$  is a smooth map, then ${\displaystyle d\phi |_{p}:T_{p}M\rightarrow T_{\phi (p)}N}$  defines a tangent map from ${\displaystyle M}$  to ${\displaystyle N}$ . It is defined through curves ${\displaystyle \gamma }$  on ${\displaystyle M}$  with derivative ${\displaystyle \gamma '(0)=X\in T_{p}M}$  such that

${\displaystyle d\phi (X):=(\phi \circ \gamma )'.}$ 

Note that ${\displaystyle \phi }$  is a ${\displaystyle 0}$ -form with values in ${\displaystyle N}$ .

Pull-back edit

If ${\displaystyle \phi :M\rightarrow N}$  is a smooth map, then the pull-back of a ${\displaystyle k}$ -form ${\displaystyle \alpha \in \Omega ^{k}(N)}$  is defined such that for any ${\displaystyle k}$ -dimensional submanifold ${\displaystyle \Sigma \subset M}$ 

${\displaystyle \int _{\Sigma }\phi ^{*}\alpha =\int _{\phi (\Sigma )}\alpha .}$ 

The pull-back can also be expressed as

${\displaystyle (\phi ^{*}\alpha )(X_{1},\ldots ,X_{k})=\alpha (d\phi (X_{1}),\ldots ,d\phi (X_{k})).}$ 

Interior product edit

Also known as the interior derivative, the interior product given a section ${\displaystyle Y\in \Gamma (TM)}$  is a map ${\displaystyle \iota _{Y}:\Omega ^{k+1}(M)\rightarrow \Omega ^{k}(M)}$  that effectively substitutes the first input of a ${\displaystyle (k+1)}$ -form with ${\displaystyle Y}$ . If ${\displaystyle \alpha \in \Omega ^{k+1}(M)}$  and ${\displaystyle X_{i}\in \Gamma (TM)}$  then

${\displaystyle (\iota _{Y}\alpha )(X_{1},\ldots ,X_{k})=\alpha (Y,X_{1},\ldots ,X_{k}).}$ 

Metric tensor edit

Given a nondegenerate bilinear form ${\displaystyle g_{p}(\cdot ,\cdot )}$  on each ${\displaystyle T_{p}M}$  that is continuous on ${\displaystyle M}$ , the manifold becomes a pseudo-Riemannian manifold. We denote the metric tensor ${\displaystyle g}$ , defined pointwise by ${\displaystyle g(X,Y)|_{p}=g_{p}(X|_{p},Y|_{p})}$ . We call ${\displaystyle s=\operatorname {sign} (g)}$  the signature of the metric. A Riemannian manifold has ${\displaystyle s=1}$ , whereas Minkowski space has ${\displaystyle s=-1}$ .

Musical isomorphisms edit

The metric tensor ${\displaystyle g(\cdot ,\cdot )}$  induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat ${\displaystyle \flat }$  and sharp ${\displaystyle \sharp }$ . A section ${\displaystyle A\in \Gamma (TM)}$  corresponds to the unique one-form ${\displaystyle A^{\flat }\in \Omega ^{1}(M)}$  such that for all sections ${\displaystyle X\in \Gamma (TM)}$ , we have:

${\displaystyle A^{\flat }(X)=g(A,X).}$ 

A one-form ${\displaystyle \alpha \in \Omega ^{1}(M)}$  corresponds to the unique vector field ${\displaystyle \alpha ^{\sharp }\in \Gamma (TM)}$  such that for all ${\displaystyle X\in \Gamma (TM)}$ , we have:

${\displaystyle \alpha (X)=g(\alpha ^{\sharp },X).}$ 

These mappings extend via multilinearity to mappings from ${\displaystyle k}$ -vector fields to ${\displaystyle k}$ -forms and ${\displaystyle k}$ -forms to ${\displaystyle k}$ -vector fields through

${\displaystyle (A_{1}\wedge A_{2}\wedge \cdots \wedge A_{k})^{\flat }=A_{1}^{\flat }\wedge A_{2}^{\flat }\wedge \cdots \wedge A_{k}^{\flat }}$ 

${\displaystyle (\alpha _{1}\wedge \alpha _{2}\wedge \cdots \wedge \alpha _{k})^{\sharp }=\alpha _{1}^{\sharp }\wedge \alpha _{2}^{\sharp }\wedge \cdots \wedge \alpha _{k}^{\sharp }.}$ 

Hodge star edit

For an n-manifold M, the Hodge star operator ${\displaystyle {\star }:\Omega ^{k}(M)\rightarrow \Omega ^{n-k}(M)}$  is a duality mapping taking a ${\displaystyle k}$ -form ${\displaystyle \alpha \in \Omega ^{k}(M)}$  to an ${\displaystyle (n{-}k)}$ -form ${\displaystyle ({\star }\alpha )\in \Omega ^{n-k}(M)}$ .

It can be defined in terms of an oriented frame ${\displaystyle (X_{1},\ldots ,X_{n})}$  for ${\displaystyle TM}$ , orthonormal with respect to the given metric tensor ${\displaystyle g}$ :

${\displaystyle ({\star }\alpha )(X_{1},\ldots ,X_{n-k})=\alpha (X_{n-k+1},\ldots ,X_{n}).}$ 

Co-differential operator edit

The co-differential operator ${\displaystyle \delta :\Omega ^{k}(M)\rightarrow \Omega ^{k-1}(M)}$  on an ${\displaystyle n}$  dimensional manifold ${\displaystyle M}$  is defined by

${\displaystyle \delta :=(-1)^{k}{\star }^{-1}d{\star }=(-1)^{nk+n+1}{\star }d{\star }.}$ 

The Hodge–Dirac operator, ${\displaystyle d+\delta }$ , is a Dirac operator studied in Clifford analysis.

Oriented manifold edit

An ${\displaystyle n}$ -dimensional orientable manifold M is a manifold that can be equipped with a choice of an n-form ${\displaystyle \mu \in \Omega ^{n}(M)}$  that is continuous and nonzero everywhere on M.

Volume form edit

On an orientable manifold ${\displaystyle M}$  the canonical choice of a volume form given a metric tensor ${\displaystyle g}$  and an orientation is ${\displaystyle \mathbf {det} :={\sqrt {|\det g|}}\;dX_{1}^{\flat }\wedge \ldots \wedge dX_{n}^{\flat }}$  for any basis ${\displaystyle dX_{1},\ldots ,dX_{n}}$  ordered to match the orientation.

Area form edit

Given a volume form ${\displaystyle \mathbf {det} }$  and a unit normal vector ${\displaystyle N}$  we can also define an area form ${\displaystyle \sigma :=\iota _{N}{\textbf {det}}}$  on the boundary ${\displaystyle \partial M.}$ 

Bilinear form on k-forms edit

A generalization of the metric tensor, the symmetric bilinear form between two ${\displaystyle k}$ -forms ${\displaystyle \alpha ,\beta \in \Omega ^{k}(M)}$ , is defined pointwise on ${\displaystyle M}$  by

${\displaystyle \langle \alpha ,\beta \rangle |_{p}:={\star }(\alpha \wedge {\star }\beta )|_{p}.}$ 

The ${\displaystyle L^{2}}$ -bilinear form for the space of ${\displaystyle k}$ -forms ${\displaystyle \Omega ^{k}(M)}$  is defined by

${\displaystyle \langle \!\langle \alpha ,\beta \rangle \!\rangle :=\int _{M}\alpha \wedge {\star }\beta .}$ 

In the case of a Riemannian manifold, each is an inner product (i.e. is positive-definite).

Lie derivative edit

We define the Lie derivative ${\displaystyle {\mathcal {L}}:\Omega ^{k}(M)\rightarrow \Omega ^{k}(M)}$  through Cartan's magic formula for a given section ${\displaystyle X\in \Gamma (TM)}$  as

${\displaystyle {\mathcal {L}}_{X}=d\circ \iota _{X}+\iota _{X}\circ d.}$ 

It describes the change of a ${\displaystyle k}$ -form along a flow ${\displaystyle \phi _{t}}$  associated to the section ${\displaystyle X}$ .

Laplace–Beltrami operator edit

The Laplacian ${\displaystyle \Delta :\Omega ^{k}(M)\rightarrow \Omega ^{k}(M)}$  is defined as ${\displaystyle \Delta =-(d\delta +\delta d)}$ .

Definitions on Ω^k(M) edit

${\displaystyle \alpha \in \Omega ^{k}(M)}$  is called...

• closed if ${\displaystyle d\alpha =0}$ 
• exact if ${\displaystyle \alpha =d\beta }$  for some ${\displaystyle \beta \in \Omega ^{k-1}}$ 
• coclosed if ${\displaystyle \delta \alpha =0}$ 
• coexact if ${\displaystyle \alpha =\delta \beta }$  for some ${\displaystyle \beta \in \Omega ^{k+1}}$ 
• harmonic if closed and coclosed

Cohomology edit

The ${\displaystyle k}$ -th cohomology of a manifold ${\displaystyle M}$  and its exterior derivative operators ${\displaystyle d_{0},\ldots ,d_{n-1}}$  is given by

${\displaystyle H^{k}(M):={\frac {{\text{ker}}(d_{k})}{{\text{im}}(d_{k-1})}}}$ 

Two closed ${\displaystyle k}$ -forms ${\displaystyle \alpha ,\beta \in \Omega ^{k}(M)}$  are in the same cohomology class if their difference is an exact form i.e.

${\displaystyle [\alpha ]=[\beta ]\ \ \Longleftrightarrow \ \ \alpha {-}\beta =d\eta \ {\text{ for some }}\eta \in \Omega ^{k-1}(M)}$ 

A closed surface of genus ${\displaystyle g}$  will have ${\displaystyle 2g}$  generators which are harmonic.

Dirichlet energy edit

Given ${\displaystyle \alpha \in \Omega ^{k}(M)}$ , its Dirichlet energy is

${\displaystyle {\mathcal {E}}_{\text{D}}(\alpha ):={\dfrac {1}{2}}\langle \!\langle d\alpha ,d\alpha \rangle \!\rangle +{\dfrac {1}{2}}\langle \!\langle \delta \alpha ,\delta \alpha \rangle \!\rangle }$ 

Exterior derivative properties edit

${\displaystyle \int _{\Sigma }d\alpha =\int _{\partial \Sigma }\alpha }$  ( Stokes' theorem )

${\displaystyle d\circ d=0}$  ( cochain complex )

${\displaystyle d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1)^{k}\alpha \wedge d\beta }$  for ${\displaystyle \alpha \in \Omega ^{k}(M),\ \beta \in \Omega ^{l}(M)}$  ( Leibniz rule )

${\displaystyle df(X)=\partial _{X}f}$  for ${\displaystyle f\in \Omega ^{0}(M),\ X\in \Gamma (TM)}$  ( directional derivative )

${\displaystyle d\alpha =0}$  for ${\displaystyle \alpha \in \Omega ^{n}(M),\ {\text{dim}}(M)=n}$ 

Exterior product properties edit

${\displaystyle \alpha \wedge \beta =(-1)^{kl}\beta \wedge \alpha }$  for ${\displaystyle \alpha \in \Omega ^{k}(M),\ \beta \in \Omega ^{l}(M)}$  ( alternating )

${\displaystyle (\alpha \wedge \beta )\wedge \gamma =\alpha \wedge (\beta \wedge \gamma )}$  ( associativity )

${\displaystyle (\lambda \alpha )\wedge \beta =\lambda (\alpha \wedge \beta )}$  for ${\displaystyle \lambda \in \mathbb {R} }$  ( compatibility of scalar multiplication )

${\displaystyle \alpha \wedge (\beta _{1}+\beta _{2})=\alpha \wedge \beta _{1}+\alpha \wedge \beta _{2}}$  ( distributivity over addition )

${\displaystyle \alpha \wedge \alpha =0}$  for ${\displaystyle \alpha \in \Omega ^{k}(M)}$  when ${\displaystyle k}$  is odd or ${\displaystyle \operatorname {rank} \alpha \leq 1}$ . The rank of a ${\displaystyle k}$ -form ${\displaystyle \alpha }$  means the minimum number of monomial terms (exterior products of one-forms) that must be summed to produce ${\displaystyle \alpha }$ .

Pull-back properties edit

${\displaystyle d(\phi ^{*}\alpha )=\phi ^{*}(d\alpha )}$  ( commutative with ${\displaystyle d}$  )

${\displaystyle \phi ^{*}(\alpha \wedge \beta )=(\phi ^{*}\alpha )\wedge (\phi ^{*}\beta )}$  ( distributes over ${\displaystyle \wedge }$  )

${\displaystyle (\phi _{1}\circ \phi _{2})^{*}=\phi _{2}^{*}\phi _{1}^{*}}$  ( contravariant )

${\displaystyle \phi ^{*}f=f\circ \phi }$  for ${\displaystyle f\in \Omega ^{0}(N)}$  ( function composition )

Musical isomorphism properties edit

${\displaystyle (X^{\flat })^{\sharp }=X}$ 

${\displaystyle (\alpha ^{\sharp })^{\flat }=\alpha }$ 

Interior product properties edit

${\displaystyle \iota _{X}\circ \iota _{X}=0}$  ( nilpotent )

${\displaystyle \iota _{X}\circ \iota _{Y}=-\iota _{Y}\circ \iota _{X}}$ 

${\displaystyle \iota _{X}(\alpha \wedge \beta )=(\iota _{X}\alpha )\wedge \beta +(-1)^{k}\alpha \wedge (\iota _{X}\beta )}$  for ${\displaystyle \alpha \in \Omega ^{k}(M),\ \beta \in \Omega ^{l}(M)}$  ( Leibniz rule )

${\displaystyle \iota _{X}\alpha =\alpha (X)}$  for ${\displaystyle \alpha \in \Omega ^{1}(M)}$ 

${\displaystyle \iota _{X}f=0}$  for ${\displaystyle f\in \Omega ^{0}(M)}$ 

${\displaystyle \iota _{X}(f\alpha )=f\iota _{X}\alpha }$  for ${\displaystyle f\in \Omega ^{0}(M)}$ 

Hodge star properties edit

${\displaystyle {\star }(\lambda _{1}\alpha +\lambda _{2}\beta )=\lambda _{1}({\star }\alpha )+\lambda _{2}({\star }\beta )}$  for ${\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {R} }$  ( linearity )

${\displaystyle {\star }{\star }\alpha =s(-1)^{k(n-k)}\alpha }$  for ${\displaystyle \alpha \in \Omega ^{k}(M)}$ , ${\displaystyle n=\dim(M)}$ , and ${\displaystyle s=\operatorname {sign} (g)}$  the sign of the metric

${\displaystyle {\star }^{(-1)}=s(-1)^{k(n-k)}{\star }}$  ( inversion )

${\displaystyle {\star }(f\alpha )=f({\star }\alpha )}$  for ${\displaystyle f\in \Omega ^{0}(M)}$  ( commutative with ${\displaystyle 0}$ -forms )

${\displaystyle \langle \!\langle \alpha ,\alpha \rangle \!\rangle =\langle \!\langle {\star }\alpha ,{\star }\alpha \rangle \!\rangle }$  for ${\displaystyle \alpha \in \Omega ^{1}(M)}$  ( Hodge star preserves ${\displaystyle 1}$ -form norm )

${\displaystyle {\star }\mathbf {1} =\mathbf {det} }$  ( Hodge dual of constant function 1 is the volume form )

Co-differential operator properties edit

${\displaystyle \delta \circ \delta =0}$  ( nilpotent )

${\displaystyle {\star }\delta =(-1)^{k}d{\star }}$  and ${\displaystyle {\star }d=(-1)^{k+1}\delta {\star }}$  ( Hodge adjoint to ${\displaystyle d}$  )

${\displaystyle \langle \!\langle d\alpha ,\beta \rangle \!\rangle =\langle \!\langle \alpha ,\delta \beta \rangle \!\rangle }$  if ${\displaystyle \partial M=0}$  ( ${\displaystyle \delta }$  adjoint to ${\displaystyle d}$  )

In general, ${\displaystyle \int _{M}d\alpha \wedge \star \beta =\int _{\partial M}\alpha \wedge \star \beta +\int _{M}\alpha \wedge \star \delta \beta }$ 

${\displaystyle \delta f=0}$  for ${\displaystyle f\in \Omega ^{0}(M)}$ 

Lie derivative properties edit

${\displaystyle d\circ {\mathcal {L}}_{X}={\mathcal {L}}_{X}\circ d}$  ( commutative with ${\displaystyle d}$  )

${\displaystyle \iota _{X}\circ {\mathcal {L}}_{X}={\mathcal {L}}_{X}\circ \iota _{X}}$  ( commutative with ${\displaystyle \iota _{X}}$  )

${\displaystyle {\mathcal {L}}_{X}(\iota _{Y}\alpha )=\iota _{[X,Y]}\alpha +\iota _{Y}{\mathcal {L}}_{X}\alpha }$ 

${\displaystyle {\mathcal {L}}_{X}(\alpha \wedge \beta )=({\mathcal {L}}_{X}\alpha )\wedge \beta +\alpha \wedge ({\mathcal {L}}_{X}\beta )}$  ( Leibniz rule )

${\displaystyle \iota _{X}({\star }\mathbf {1} )={\star }X^{\flat }}$ 

${\displaystyle \iota _{X}({\star }\alpha )=(-1)^{k}{\star }(X^{\flat }\wedge \alpha )}$  if ${\displaystyle \alpha \in \Omega ^{k}(M)}$ 

${\displaystyle \iota _{X}(\phi ^{*}\alpha )=\phi ^{*}(\iota _{d\phi (X)}\alpha )}$ 

${\displaystyle \nu ,\mu \in \Omega ^{n}(M),\mu {\text{ non-zero }}\ \Rightarrow \ \exists \ f\in \Omega ^{0}(M):\ \nu =f\mu }$ 

${\displaystyle X^{\flat }\wedge {\star }Y^{\flat }=g(X,Y)({\star }\mathbf {1} )}$  ( bilinear form )

${\displaystyle [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0}$  ( Jacobi identity )

Dimensions edit

If ${\displaystyle n=\dim M}$ 

${\displaystyle \dim \Omega ^{k}(M)={\binom {n}{k}}}$  for ${\displaystyle 0\leq k\leq n}$ 

${\displaystyle \dim \Omega ^{k}(M)=0}$  for ${\displaystyle k<0,\ k>n}$ 

If ${\displaystyle X_{1},\ldots ,X_{n}\in \Gamma (TM)}$  is a basis, then a basis of ${\displaystyle \Omega ^{k}(M)}$  is

${\displaystyle \{X_{\sigma (1)}^{\flat }\wedge \ldots \wedge X_{\sigma (k)}^{\flat }\ :\ \sigma \in S(k,n)\}}$ 

Exterior products edit

Let ${\displaystyle \alpha ,\beta ,\gamma ,\alpha _{i}\in \Omega ^{1}(M)}$  and ${\displaystyle X,Y,Z,X_{i}}$  be vector fields.

${\displaystyle \alpha (X)=\det {\begin{bmatrix}\alpha (X)\\\end{bmatrix}}}$ 

${\displaystyle (\alpha \wedge \beta )(X,Y)=\det {\begin{bmatrix}\alpha (X)&\alpha (Y)\\\beta (X)&\beta (Y)\\\end{bmatrix}}}$ 

${\displaystyle (\alpha \wedge \beta \wedge \gamma )(X,Y,Z)=\det {\begin{bmatrix}\alpha (X)&\alpha (Y)&\alpha (Z)\\\beta (X)&\beta (Y)&\beta (Z)\\\gamma (X)&\gamma (Y)&\gamma (Z)\end{bmatrix}}}$ 

${\displaystyle (\alpha _{1}\wedge \ldots \wedge \alpha _{l})(X_{1},\ldots ,X_{l})=\det {\begin{bmatrix}\alpha _{1}(X_{1})&\alpha _{1}(X_{2})&\dots &\alpha _{1}(X_{l})\\\alpha _{2}(X_{1})&\alpha _{2}(X_{2})&\dots &\alpha _{2}(X_{l})\\\vdots &\vdots &\ddots &\vdots \\\alpha _{l}(X_{1})&\alpha _{l}(X_{2})&\dots &\alpha _{l}(X_{l})\end{bmatrix}}}$ 

Projection and rejection edit

${\displaystyle (-1)^{k}\iota _{X}{\star }\alpha ={\star }(X^{\flat }\wedge \alpha )}$  ( interior product ${\displaystyle \iota _{X}{\star }}$  dual to wedge ${\displaystyle X^{\flat }\wedge }$  )

${\displaystyle (\iota _{X}\alpha )\wedge {\star }\beta =\alpha \wedge {\star }(X^{\flat }\wedge \beta )}$  for ${\displaystyle \alpha \in \Omega ^{k+1}(M),\beta \in \Omega ^{k}(M)}$ 

If ${\displaystyle |X|=1,\ \alpha \in \Omega ^{k}(M)}$ , then

• ${\displaystyle \iota _{X}\circ (X^{\flat }\wedge ):\Omega ^{k}(M)\rightarrow \Omega ^{k}(M)}$  is the projection of ${\displaystyle \alpha }$  onto the orthogonal complement of ${\displaystyle X}$ .
• ${\displaystyle (X^{\flat }\wedge )\circ \iota _{X}:\Omega ^{k}(M)\rightarrow \Omega ^{k}(M)}$  is the rejection of ${\displaystyle \alpha }$ , the remainder of the projection.
• thus ${\displaystyle \iota _{X}\circ (X^{\flat }\wedge )+(X^{\flat }\wedge )\circ \iota _{X}={\text{id}}}$  ( projection–rejection decomposition )

Given the boundary ${\displaystyle \partial M}$  with unit normal vector ${\displaystyle N}$ 

• ${\displaystyle \mathbf {t} :=\iota _{N}\circ (N^{\flat }\wedge )}$  extracts the tangential component of the boundary.
• ${\displaystyle \mathbf {n} :=({\text{id}}-\mathbf {t} )}$  extracts the normal component of the boundary.

Sum expressions edit

${\displaystyle (d\alpha )(X_{0},\ldots ,X_{k})=\sum _{0\leq j\leq k}(-1)^{j}d(\alpha (X_{0},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}))(X_{j})+\sum _{0\leq i<j\leq k}(-1)^{i+j}\alpha ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k})}$ 

${\displaystyle (d\alpha )(X_{1},\ldots ,X_{k})=\sum _{i=1}^{k}(-1)^{i+1}(\nabla _{X_{i}}\alpha )(X_{1},\ldots ,{\hat {X}}_{i},\ldots ,X_{k})}$ 

${\displaystyle (\delta \alpha )(X_{1},\ldots ,X_{k-1})=-\sum _{i=1}^{n}(\iota _{E_{i}}(\nabla _{E_{i}}\alpha ))(X_{1},\ldots ,{\hat {X}}_{i},\ldots ,X_{k})}$  given a positively oriented orthonormal frame ${\displaystyle E_{1},\ldots ,E_{n}}$ .

${\displaystyle ({\mathcal {L}}_{Y}\alpha )(X_{1},\ldots ,X_{k})=(\nabla _{Y}\alpha )(X_{1},\ldots ,X_{k})-\sum _{i=1}^{k}\alpha (X_{1},\ldots ,\nabla _{X_{i}}Y,\ldots ,X_{k})}$ 

Hodge decomposition edit

If ${\displaystyle \partial M=\emptyset }$ , ${\displaystyle \omega \in \Omega ^{k}(M)\Rightarrow \exists \alpha \in \Omega ^{k-1},\ \beta \in \Omega ^{k+1},\ \gamma \in \Omega ^{k}(M),\ d\gamma =0,\ \delta \gamma =0}$  such that^{[citation needed]}

${\displaystyle \omega =d\alpha +\delta \beta +\gamma }$ 

Poincaré lemma edit

If a boundaryless manifold ${\displaystyle M}$  has trivial cohomology ${\displaystyle H^{k}(M)=\{0\}}$ , then any closed ${\displaystyle \omega \in \Omega ^{k}(M)}$  is exact. This is the case if M is contractible.

Identities in Euclidean 3-space edit

Let Euclidean metric ${\displaystyle g(X,Y):=\langle X,Y\rangle =X\cdot Y}$ .

We use ${\displaystyle \nabla =\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)}$  differential operator ${\displaystyle \mathbb {R} ^{3}}$ 

${\displaystyle \iota _{X}\alpha =g(X,\alpha ^{\sharp })=X\cdot \alpha ^{\sharp }}$  for ${\displaystyle \alpha \in \Omega ^{1}(M)}$ .

${\displaystyle \mathbf {det} (X,Y,Z)=\langle X,Y\times Z\rangle =\langle X\times Y,Z\rangle }$  ( scalar triple product )

${\displaystyle X\times Y=({\star }(X^{\flat }\wedge Y^{\flat }))^{\sharp }}$  ( cross product )

${\displaystyle \iota _{X}\alpha =-(X\times A)^{\flat }}$  if ${\displaystyle \alpha \in \Omega ^{2}(M),\ A=({\star }\alpha )^{\sharp }}$ 

${\displaystyle X\cdot Y={\star }(X^{\flat }\wedge {\star }Y^{\flat })}$  ( scalar product )

${\displaystyle \nabla f=(df)^{\sharp }}$  ( gradient )

${\displaystyle X\cdot \nabla f=df(X)}$  ( directional derivative )

${\displaystyle \nabla \cdot X={\star }d{\star }X^{\flat }=-\delta X^{\flat }}$  ( divergence )

${\displaystyle \nabla \times X=({\star }dX^{\flat })^{\sharp }}$  ( curl )

${\displaystyle \langle X,N\rangle \sigma ={\star }X^{\flat }}$  where ${\displaystyle N}$  is the unit normal vector of ${\displaystyle \partial M}$  and ${\displaystyle \sigma =\iota _{N}\mathbf {det} }$  is the area form on ${\displaystyle \partial M}$ .

${\displaystyle \int _{\Sigma }d{\star }X^{\flat }=\int _{\partial \Sigma }{\star }X^{\flat }=\int _{\partial \Sigma }\langle X,N\rangle \sigma }$  ( divergence theorem )

Lie derivatives edit

${\displaystyle {\mathcal {L}}_{X}f=X\cdot \nabla f}$  ( ${\displaystyle 0}$ -forms )

${\displaystyle {\mathcal {L}}_{X}\alpha =(\nabla _{X}\alpha ^{\sharp })^{\flat }+g(\alpha ^{\sharp },\nabla X)}$  ( ${\displaystyle 1}$ -forms )

${\displaystyle {\star }{\mathcal {L}}_{X}\beta =\left(\nabla _{X}B-\nabla _{B}X+({\text{div}}X)B\right)^{\flat }}$  if ${\displaystyle B=({\star }\beta )^{\sharp }}$  ( ${\displaystyle 2}$ -forms on ${\displaystyle 3}$ -manifolds )

${\displaystyle {\star }{\mathcal {L}}_{X}\rho =dq(X)+({\text{div}}X)q}$  if ${\displaystyle \rho ={\star }q\in \Omega ^{0}(M)}$  ( ${\displaystyle n}$ -forms )

${\displaystyle {\mathcal {L}}_{X}(\mathbf {det} )=({\text{div}}(X))\mathbf {det} }$