Chern–Weil Theory and Characteristic Forms

Let $G$ be either a real matrix Lie group $G\subseteq {\mathrm{GL}}_n(\mathbb{R})$ with real Lie algebra $\mathfrak{g}$, or a complex matrix Lie group $G\subseteq {\mathrm{GL}}_n(\mathbb{C})$ with complex Lie algebra $\mathfrak{g}$. In the real case let $K\in \{\mathbb{R},\mathbb{C}\}$ and use polynomials in real linear coordinates with coefficients in $K$. In the complex case take $K=\mathbb{C}$ and use polynomials in complex linear coordinates (with no conjugate-coordinate variables). A homogeneous degree-$k$ polynomial $P:\mathfrak{g}\to K$ is $G$-invariant when $P({\mathrm{Ad}}_{g}A)=P(A)(g\in G,A\in \mathfrak{g}),$ where the adjoint action is the one in Conjugation and the adjoint representation of a Lie group. For $k\ge 1$, its polarization is the unique symmetric multilinear map $P_k:{\mathfrak{g}}^k\to K$ with $P(A)=P_k(A,\dots ,A)\mathrm{.}$ Here multilinearity is over $\mathbb{R}$ when $\mathfrak{g}$ is a real Lie algebra and over $\mathbb{C}$ when it is a complex Lie algebra; thus a $\mathbb{C}$-valued polynomial on a real Lie algebra still has a real-multilinear polarization. A complex matrix Lie group may also be regarded as a real Lie group, in which case the underlying real Lie algebra and real-multilinear convention apply; this is also the convention for real-valued coordinate polynomials on a complex matrix Lie algebra, such as $z\mapsto \mathrm{Re}z$ on ${\mathfrak{g}\mathfrak{l}}_1(\mathbb{C})$.

The polarization can be computed by $P_k(A_1,\dots ,A_k)=\frac{1}{k!}[t_1\cdots t_k]P(t_1{A}_1+\cdots +t_k{A}_k),$ where the bracket extracts the coefficient of $t_1\cdots t_k$. A homogeneous degree-$k$ coordinate polynomial makes this coefficient symmetric and multilinear in the $A_j$; setting every $A_j=A$ gives the coefficient $k!P(A)$, and the same extraction proves uniqueness. Since $K$ has characteristic zero, division by $k!$ is valid. For $k=0$, the invariant polynomials are constants, viewed as symmetric $0$-linear forms $P_0\in K$. Finite sums of these homogeneous invariant polynomials form the invariant polynomial algebra; sums and products remain invariant because the adjoint action respects addition and multiplication of scalar values.

Polarization preserves invariance: simultaneous application of ${\mathrm{Ad}}_g$ to the arguments leaves every term in the coefficient formula unchanged. Conversely, if a symmetric $P_k$ is invariant under simultaneous adjoint action, its diagonal $A\mapsto P_k(A,\dots ,A)$ is a $G$-invariant polynomial. Differentiating that multilinear invariance along $g(t)=\exp (tX)$ gives the infinitesimal identity ${\sum }_{j=1}^k{P}_k(A_1,\dots ,[X,A_j],\dots ,A_k)=0(X\in \mathfrak{g})\mathrm{.}$ Indeed, for matrices, ${\frac{d}{dt}\mid }_0{\mathrm{Ad}}_{\exp (tX)}{A}_j=[X,A_j]$, so the chain rule gives exactly the displayed sum. For $k=0$ the sum is empty and equals zero.

For ${\mathfrak{g}\mathfrak{l}}_r(\mathbb{C})$, every coefficient of $\det (I-tA)$ is invariant under ${\mathrm{GL}}_r(\mathbb{C})$, since $I-t(gA{g}^{-1})=g(I-tA)g^{-1}$ and determinant is unchanged by conjugation. Its restriction to $\mathfrak{u}(r)$ is therefore invariant for the adjoint action of $U(r)$. For the Pfaffian, use only $\mathfrak{s}\mathfrak{o}(2m)$ with $G=SO(2m)$ and a fixed oriented orthonormal frame. For $A=(a_{ij})\in \mathfrak{s}\mathfrak{o}(2m)$, define ${\alpha }_A=\frac{1}{2}{\sum }_{i,j}{a}_{ij}{e}^i\wedge e^j$ and set $\frac{{\alpha }_A^m}{m!}=\mathrm{Pf}(A)e^1\wedge \cdots \wedge e^{2m}\mathrm{.}$ This is a homogeneous polynomial of degree $m$. If $S\in O(2m)$, the induced action on the top exterior power multiplies the oriented volume by $\det S$; hence $\mathrm{Pf}(SA{S}^{\mathsf{T}})=(\det S)\mathrm{Pf}(A)$. It is invariant under $SO(2m)$, and a reversal of orientation changes its sign. The convention gives $\mathrm{Pf}\text{\hspace{0.17em}⁣}\left(\begin{array}{cc}0& a\\ -a& 0\end{array}\right)=a$; for $m=0$ it gives the empty-matrix Pfaffian $1$.

Let $E\to M$ be a smooth real vector bundle of rank $2r$, where $r\ge 0$, over a smooth manifold that may have boundary. A smooth complex structure on $E$ is a smooth bundle endomorphism $J:E\to E$ with $J^2=-I$. It makes each fiber a complex vector space by $(a+ib)v=av+bJv$. Equivalently, $E$ has local smooth complex frames with transition maps in ${\mathrm{GL}}_r(\mathbb{C})$. In one direction, multiplication by $i$ in complex bundle charts gives such a smooth $J$ because the transition maps are complex-linear. Conversely, near any point choose a complex basis in its fiber and extend its vectors to local smooth real sections; those sections together with their $J$-images remain a real frame after shrinking the neighborhood, and give a local complex frame. Forgetting smoothness in these charts gives the associated rank-$r$ complex topological vector bundle of Real and complex topological vector bundles.

For a smooth complex bundle $(E,J)$, write ${\mathrm{\Gamma }}_{\mathbb{C}}(E)$ for its smooth complex sections. A complex connection is a covariant derivative $\mathrm{\nabla }:{\mathrm{\Gamma }}_{\mathbb{C}}(E)\to {\mathrm{\Omega }}^1(M;E)$ that is $\mathbb{C}$-linear and satisfies $\mathrm{\nabla }(fs)=df\otimes s+f\mathrm{\nabla }s$ for every smooth complex-valued function $f$ and section $s$. This extends the real bundle-connection convention of Connection on a smooth vector bundle. A complex connection is Hermitian for a supplied Hermitian metric $h$, taken linear in its first variable and conjugate-linear in its second, when $Xh(s,t)=h({\mathrm{\nabla }}_{X}s,t)+h(s,{\mathrm{\nabla }}_{X}t)$ for all smooth sections $s,t$ and vector fields $X$. A real connection on a Euclidean vector bundle is Euclidean-compatible when it obeys the same identity for the real bundle metric, as in Metric compatible connection on a riemannian vector bundle.

In a local complex frame $e=(e_1,\dots ,e_r)$ write $\mathrm{\nabla }{e}_j={\sum }_i{e}_i{\omega }^i{}_j$. Applying Hermitian compatibility to the frame sections gives $dH={\omega }^{\mathsf{T}}H+H\bar{\omega },H_{ij}=h(e_i,e_j)\mathrm{.}$ After smooth Gram–Schmidt, a local unitary frame has $H=I$, so ${\omega }^{\ast }=-\omega$. In a real orthonormal frame the corresponding equation is ${\omega }^{\mathsf{T}}=-\omega$. With the curvature convention $\mathrm{\Omega }=d\omega +\omega \wedge \omega$ of Curvature two-form structure equation, these identities imply ${\mathrm{\Omega }}^{\ast }=-\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\mathsf{T}}=-\mathrm{\Omega }$, respectively: exterior differentiation preserves the adjoint relation, and for a matrix of one-forms $(\omega \wedge \omega {)}^{\ast }=-\omega \wedge \omega$ because transposition reverses the matrix order while one-forms anticommute. The published local structure-equation calculation is coefficientwise, so it also applies to complex frame coefficients. Thus curvature matrices of Hermitian or Euclidean-compatible connections are skew-Hermitian or skew-symmetric in the corresponding frames. For rank zero these frame identities are vacuous. In boundary charts the same identities hold up to the boundary by restriction of the smooth half-space coefficient formulas.

Assume the Axiom of Choice (AC). Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, with boundary allowed. Every finite-rank smooth real vector bundle over $M$ admits a smooth Euclidean bundle metric and a Euclidean-compatible smooth connection. Every finite-rank smooth complex vector bundle over $M$ admits a smooth Hermitian metric and a Hermitian complex connection. In particular, any supplied Euclidean or Hermitian metric on such a bundle admits a compatible connection. Here “complex connection,” “Hermitian,” and “Euclidean-compatible” have the meanings in Complex-linear and metric-compatible bundle connections.

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, possibly with boundary. Let $\mathbb{F},\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, let $G$ be a real or complex matrix Lie group in ${\mathrm{GL}}_r(\mathbb{F})$, and let $\mathfrak{g}$ be its Lie algebra. For a smooth rank-$r$ $\mathbb{F}$-vector bundle $E\to M$, a $G$-frame atlas is a supplied open cover with local frames $e_{\alpha }$ whose transition matrices $A_{\beta \alpha }$, defined by $e_{\beta }=e_{\alpha }{A}_{\beta \alpha }$, take values in $G$. This is the local-frame description of a reduction of the frame group to $G$. If $\mathbb{F}=\mathbb{C}$, a connection means a complex connection as in Complex-linear and metric-compatible bundle connections; for $\mathbb{F}=\mathbb{R}$ it means an ordinary smooth bundle connection. A connection is compatible with this reduction when its connection matrix ${\omega }_{\alpha }$ in every supplied $G$-frame is $\mathfrak{g}$-valued. In the convention of Curvature two-form structure equation, its curvature matrix is

It is $\mathfrak{g}$-valued: for tangent vectors $X,Y$, the second term evaluates to $[{\omega }_{\alpha }(X),{\omega }_{\alpha }(Y)]$, and both this bracket and $d{\omega }_{\alpha }(X,Y)$ lie in $\mathfrak{g}$.

Let $P:\mathfrak{g}\to \mathbb{K}$ be a homogeneous degree-$k$ $G$-invariant polynomial, with symmetric multilinear polarization $P_k$ as in Invariant symmetric polynomials on a matrix Lie algebra. For $k\ge 1$, define its evaluation on curvature in a supplied $G$-frame by the alternating $2k$-form

where $(T_a)$ is any basis of $\mathfrak{g}$ and ${\mathrm{\Omega }}_{\alpha }={\sum }_a{T}_a\otimes {\mathrm{\Omega }}_{\alpha }^a$. The right side is the multilinear extension of $P_k$ followed by exterior multiplication, so it is independent of the chosen basis or tensor decomposition. Equivalently, for $v_1,\dots ,v_{2k}\in T_{x}M$,

For $k=0$, set $P_0({\mathrm{\Omega }}_{\alpha }^0)=P_0$, the corresponding constant $\mathbb{K}$-valued $0$-form. A finite sum of homogeneous invariant polynomials is evaluated degree by degree and the resulting forms are added.

These local forms agree on overlaps. Indeed, Vector-bundle curvature is an endomorphism-valued two-form makes the curvature a global $\mathrm{End}(E)$-valued $2$-form. To see its frame transformation, if a fibre vector has coordinate columns $x_{\alpha }=A_{\beta \alpha }{x}_{\beta }$ and an endomorphism has matrices $T_{\alpha },T_{\beta }$, then $T_{\beta }=A_{\beta \alpha }^{-1}{T}_{\alpha }{A}_{\beta \alpha }$. Applying this pointwise to curvature gives ${\mathrm{\Omega }}_{\beta }=A_{\beta \alpha }^{-1}{\mathrm{\Omega }}_{\alpha }{A}_{\beta \alpha }$. Since $A_{\beta \alpha }\in G$, simultaneous $G$-invariance of $P_k$ gives

Thus the local evaluations define a global $\mathbb{K}$-valued $2k$-form. For $\mathbb{K}=\mathbb{C}$, this means a complex-valued form, obtained by complexifying the real form convention; the same local formulas are smooth up to boundary charts by The de Rham complex and pullback extend to manifolds with boundary.

All scalar $2$-form coefficients commute under exterior multiplication because their degree is even. Matrix factors inside polynomial expressions, including traces and determinant coefficients, retain the order prescribed by the matrix polynomial.

The reduction and compatible connection are part of the input when $P$ is invariant only under $G$. For a polynomial invariant under the full general linear group, the construction may use the full frame group. The in-scope applications use ${\mathrm{GL}}_r(\mathbb{C})$-invariant polynomials for Chern forms, their complexified analogues for Pontryagin forms, and the Pfaffian on $\mathfrak{s}\mathfrak{o}(2m)$ with an oriented orthonormal frame for Euler forms.

The local $G$-frames and compatible connection are supplied data. The definition makes no simultaneous global choice of frames, and its construction uses no axiom of choice.

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, possibly with boundary. Let $G$ be a real or complex matrix Lie group with Lie algebra $\mathfrak{g}$, and let $P_k:{\mathfrak{g}}^k\to \mathbb{K}$ be the symmetric multilinear polarization of a homogeneous $G$-invariant polynomial, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Work in a supplied $G$-frame chart with a connection compatible with that reduction, so its local connection form $\omega$ is $\mathfrak{g}$-valued. For each homogeneous $\mathfrak{g}$-valued form $A_j\in {\mathrm{\Omega }}^{q_j}(U;\mathfrak{g})$, extend $P_k$ by applying it to the Lie-algebra coefficients and wedging the scalar-form coefficients in the displayed argument order. Define $D^{\mathrm{\nabla }}A=dA+\omega \wedge A-(-1{)}^{q}A\wedge \omega (A\in {\mathrm{\Omega }}^q(U;\mathfrak{g}))\mathrm{.}$ Then $d{P}_k(A_1,\dots ,A_k)={\sum }_{j=1}^k(-1{)}^{q_1+\cdots +q_{j-1}}{P}_k(A_1,\dots ,D^{\mathrm{\nabla }}{A}_j,\dots ,A_k)\mathrm{.}$ In particular, the signed sum of the graded connection-commutator terms is zero. For $k=0$, the assertion is $d{P}_0=0$. The identity is local and hence also holds in boundary charts by restriction of the same coefficient calculation.

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, possibly with boundary. Let $E\to M$ be a smooth finite-rank real or complex vector bundle with a supplied $G$-frame atlas and a connection compatible with that reduction, and let $P_k$ be a homogeneous degree-$k$ $G$-invariant polynomial as in Evaluation of an invariant polynomial on curvature. If $\mathrm{\Omega }$ is the curvature matrix in a supplied $G$-frame, then its global evaluation $P_k(\mathrm{\Omega },\dots ,\mathrm{\Omega })$ is closed. This includes $k=0$, where the evaluation is the constant $0$-form. The assertion applies to $G={\mathrm{GL}}_r(\mathbb{C})$, ${\mathrm{GL}}_r(\mathbb{R})$, $U(r)$, and $\mathrm{SO}(2m)$ within their stated invariant-polynomial scopes. Finite sums are closed degree by degree.

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, possibly with boundary. Let $E\to M$ be a smooth rank-$r$ real or complex vector bundle with a supplied $G$-frame atlas, where $G$ is a real or complex matrix Lie group, and let $\mathrm{\nabla }$ be a fixed connection compatible with that reduction. For the algebra of finite sums of $\mathrm{Ad}(G)$-invariant polynomials $P={\sum }_k{P}_k$ with values in $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. For each $P_k$, write $P_k^{\mathrm{p}\mathrm{o}\mathrm{l}}$ for its normalized symmetric polarization (with $P_0^{\mathrm{p}\mathrm{o}\mathrm{l}}=P_0$), and define ${\mathrm{CW}}_{E,\mathrm{\nabla }}(P)={\sum }_k[P_k^{\mathrm{p}\mathrm{o}\mathrm{l}}({\mathrm{\Omega }}_{\mathrm{\nabla }},\dots ,{\mathrm{\Omega }}_{\mathrm{\nabla }})]\in H_{\mathrm{d}\mathrm{R}}^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}(M;\mathbb{K})\mathrm{.}$ where a degree-$k$ polynomial maps to cohomological degree $2k$. The map is a graded unital $\mathbb{K}$-algebra homomorphism, with polynomial multiplication on the source and wedge product on the target, and ${\mathrm{CW}}_{E,\mathrm{\nabla }}(1)=[1]$. The coefficient convention for $H_{\mathrm{d}\mathrm{R}}(M;\mathbb{K})$ on boundary manifolds is specified below. This definition depends on the fixed connection; independence of its cohomology value is a later theorem.

For the real coefficient target, use the ordinary real de Rham complex when $\mathrm{\partial }M=\mathrm{\varnothing }$ and the locally extendible boundary-chart complex from [F5] when $M$ has boundary; write its cohomology ring as $H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{R})$. Its product is induced by wedge: the graded Leibniz rule in [F5] makes exact changes of a closed representative exact. For $\mathbb{K}=\mathbb{R}$, set $H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{K})=H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{R})$. For $\mathbb{K}=\mathbb{C}$, set ${\mathrm{\Omega }}^{\bullet }(M;\mathbb{C})={\mathrm{\Omega }}^{\bullet }(M;\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C},d_{\mathbb{C}}=d_{\mathbb{R}}\otimes 1,H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{C})=H^{\bullet }({\mathrm{\Omega }}^{\bullet }(M;\mathbb{C}),d_{\mathbb{C}})\mathrm{.}$ This is the complex-valued smooth-form complex, with boundary coefficients locally extendible componentwise. Real and imaginary parts split its cycles and exact forms, so its cohomology is $H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C}$; wedge and the unit extend $\mathbb{C}$-linearly. On the empty manifold the target is the zero algebra with $1=0$, as in [F7].

For a polynomial $P_k$ homogeneous of degree $k$, use its polarization $P_k^{\mathrm{p}\mathrm{o}\mathrm{l}}$ in the global form from [F2]. Define the map by taking its cohomology class and summing over the homogeneous components. No connection-independence assertion is included in the definition.

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, possibly with boundary. Let $E\to M$ be a finite-rank real or complex smooth vector bundle with a fixed $G$-frame reduction, and let ${\mathrm{\nabla }}_0,{\mathrm{\nabla }}_1$ be connections compatible with that same reduction. Put $A={\mathrm{\nabla }}_1-{\mathrm{\nabla }}_0$, ${\mathrm{\nabla }}_t={\mathrm{\nabla }}_0+tA$, and let ${\mathrm{\Omega }}_t$ be the curvature of ${\mathrm{\nabla }}_t$. For a homogeneous degree-$k$ $G$-invariant polynomial $P_k$ with symmetric polarization, $k\ge 1$, define

This is a global $(2k-1)$-form and

For $k=0$, the two constant curvature evaluations agree, so their difference is zero. The result applies to the ${\mathrm{GL}}_r(\mathbb{C})$, ${\mathrm{GL}}_r(\mathbb{R})$, $U(r)$, and $\mathrm{SO}(2m)$ reductions with their respective invariant polynomials.

Let $M$ and $N$ be finite-dimensional Hausdorff second-countable smooth manifolds, with boundary allowed. Let $E\to M$ be a finite-rank real or complex bundle with a supplied $G$-frame reduction, where the applicable group is ${\mathrm{GL}}_r(\mathbb{R})$, ${\mathrm{GL}}_r(\mathbb{C})$, $U(r)$, or $SO(r)$. For a finite sum $P$ of $G$-invariant polynomials, let ${\mathrm{CW}}_{E,\mathrm{\nabla }}(P)$ be the map defined in Chern–Weil map for a chosen connection using a connection compatible with this same reduction.

For any two compatible connections ${\mathrm{\nabla }}_0,{\mathrm{\nabla }}_1$ on $E$, ${\mathrm{CW}}_{E,{\mathrm{\nabla }}_0}(P)={\mathrm{CW}}_{E,{\mathrm{\nabla }}_1}(P)\mathrm{.}$ For every smooth map $f:N\to M$, equip $f^{\ast }E$ with its pulled-back $G$-reduction and use the pullback connection $f^{\ast }\mathrm{\nabla }$. Then ${\mathrm{CW}}_{f^{\ast }E,f^{\ast }\mathrm{\nabla }}(P)=f^{\ast }{\mathrm{CW}}_{E,\mathrm{\nabla }}(P),$ where for boundary manifolds smoothness and pullback of forms use the library's local-extension convention. For each supplied compatible connection, ${\mathrm{CW}}_{E,\mathrm{\nabla }}$ is a unital graded algebra homomorphism in $P$.

Assume full Axiom of Choice (AC) for the existence assertion: every such real or complex bundle with a fixed $\mathrm{GL}$, $U$, or $SO$ reduction admits a compatible connection. The connection-independence and pullback assertions for supplied compatible connections do not use AC.

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, possibly with boundary. If $E\to M$ is a rank-$r$ complex vector bundle with complex connection $\mathrm{\nabla }$ and curvature $\mathrm{\Omega }$, define the total Chern form $c(\mathrm{\nabla })=\det \text{\hspace{0.17em}⁣}(I-\frac{\mathrm{\Omega }}{2\pi i})={\sum }_{j=0}^r{c}_j(\mathrm{\nabla }),$ where $c_j(\mathrm{\nabla })$ has degree $2j$. For a Hermitian connection these forms are real-valued; a general complex connection need not give real-valued forms.

For a rank-$r$ real vector bundle with real connection $\mathrm{\nabla }$, let $E_{\mathbb{C}}=E{\otimes }_{\mathbb{R}}\mathbb{C}$ carry the complexified connection ${\mathrm{\nabla }}_{\mathbb{C}}$ and set $p_j(\mathrm{\nabla })=(-1{)}^j{c}_{2j}({\mathrm{\nabla }}_{\mathbb{C}})\text{in degree }4j,p(\mathrm{\nabla })={\sum }_{j\ge 0}{p}_j(\mathrm{\nabla })\mathrm{.}$ Then $p_0=1$, $p_j=0$ when $2j>r$, and every $p_j(\mathrm{\nabla })$ is real-valued, even if $\mathrm{\nabla }$ is not metric-compatible. If $\mathrm{\nabla }$ is compatible with a Euclidean metric, each odd Chern form $c_{2j+1}({\mathrm{\nabla }}_{\mathbb{C}})$ vanishes pointwise. Assume full Axiom of Choice (AC); for every real connection each $c_{2j+1}({\mathrm{\nabla }}_{\mathbb{C}})$ is then exact, with AC used through existence of a metric-compatible connection and the transgression lemma.

If $E$ is an oriented Euclidean bundle of even rank $2m$ and $\mathrm{\nabla }$ is metric-compatible, define $e(\mathrm{\nabla })=\mathrm{Pf}\text{\hspace{0.17em}⁣}\left(\frac{\mathrm{\Omega }}{2\pi }\right),$ where the Pfaffian uses the ordered oriented orthonormal frame and $\mathrm{Pf}\text{\hspace{0.17em}⁣}\left(\begin{array}{cc}0& a\\ -a& 0\end{array}\right)=a$. These curvature evaluations are closed forms. In rank zero, $c(\mathrm{\nabla })=p(\mathrm{\nabla })=e(\mathrm{\nabla })=1$. The Euler form is defined here only for oriented even-rank Euclidean bundles with a metric-compatible connection.