用户: Eee/Chern-Weil

Mostly following Morita.

1Laplacian and Harmonic forms

  In this chapter, we will see the more precise structure of the de Rham cohomology group of $M$.

  We may prove that within the set of all closed forms representing a de Rham cohomology class, there is one and only one differential form that has the best shape.

Defition 1.1. We define an inner product of forms on a single verctor space.

  For two elements of the form ${\alpha }_1\wedge \cdots \wedge {\alpha }_k$ and ${\beta }_1\wedge \cdots \wedge {\beta }_k({\alpha }_i,{\beta }_j\in V^{\ast })$, we define the value of their inner product to be $⟨{\alpha }_1\wedge \cdots \wedge {\alpha }_k,{\beta }_1\wedge \cdots \wedge {\beta }_k⟩=\det (({\alpha }_i,{\beta }_j))$.

  That this value is independent of the way the two elements are represented follows from the properties of exterior product and determinate. We now extend the inner product so defined to the whole space ${\Lambda }^k{V}^{\ast }$ by linearity.

  If $e_1,\cdots ,e_n$ is an orthonormal basis of $V$ and ${\theta }_1,\cdots ,{\theta }_n$ the dual basis, then all the elements of the form $${\theta }_{i_1}\wedge \cdots \wedge {\theta }_{i_k},1\le i_1<\cdots <i_k\le n,$$form an orthonomal basis of ${\Lambda }^k{V}^{\ast }$. From this point of view, ${\Lambda }^k{V}^{\ast }$ is of finite dimension.

Remark. In the definition of $⟨{\alpha }_1\wedge \cdots \wedge {\alpha }_k,{\beta }_1\wedge \cdots \wedge {\beta }_k⟩=\det (({\alpha }_i,{\beta }_j))$, we identify the cotangent vectors and tangent vectors, that is $$\begin{array}{rl}⟨{\alpha }_1\wedge \cdots \wedge {\alpha }_k,{\beta }_1\wedge \cdots \wedge {\beta }_k⟩& ={\alpha }_1\wedge \cdots \wedge {\alpha }_k({\beta }_1^{\ast },\cdots ,{\beta }_k^{\ast })\\ & =\det ({\alpha }_i({\beta }_j^{\ast }))\\ & =\det (({\alpha }_i,{\beta }_j)),\end{array}$$where the ${\beta }_j^{\ast }$ denote the cotangent vector of ${\beta }_j$.

Defition 1.2. We define an inner product of forms on manifold, it’s a function.

  For any two $k-$forms $\omega$ and $\eta$ on $M$, we have the inner prodcut $⟨{\omega }_p,{\eta }_p⟩$ for each $p\in M$, and thus the function $⟨\omega ,\eta ⟩$ on $M$.

  For the case $k=0$, we define the inner product of two forms, namely functions, to be their product. We also define the inner product between two differential forms of different degrees to be $0$.

  Recall that we use ${\mathcal{A}}^k(M)$ to represent the $\mathbb{R}-$linear space of all the $k-$forms on $M$. Note that this linear space is of infinite dimension.

Defition 1.3. Hodge star operator

  On an oriented Riemannian manifold $M^n$, for any integer $k(0\le k\le n)$, ${\Lambda }^k{T}_p^{\ast }M$ and ${\Lambda }^{n-k}{T}_p^{\ast }M$ have the same dimension as vector spaces, and so they are isomorphic.

  Thus we have the natural isomorphic $$\begin{array}{rl}\ast :{\Lambda }^k{T}_p^{\ast }M& \stackrel{\cong }{\to }{\Lambda }^{n-k}{T}_p^{\ast }M\\ {\theta }_1\wedge \cdots \wedge {\theta }_k& \mapsto {\theta }_{k+1}\wedge \cdots \wedge {\theta }_n,\end{array}$$where $\{{\theta }_i\}$ is an orthonormal basis of $T_p^{\ast }M$.

  Now we define $\ast$-operator: ${\mathcal{A}}^k(M)\to {\mathcal{A}}^{n-k}(M)$ globally on $M$:

  Locally, we choose $e_1,\cdots ,e_n$ to be an orthonomal frame, and ${\theta }_1,\cdots ,{\theta }_n$ to be the dual basis frame.

  For $\omega =\sum _{i_1<\cdots <i_k}{f}_{i_1\cdots i_k}{\theta }_{i_1}\wedge \cdots \wedge {\theta }_{i_k}$, we have $$\ast \omega =\sum _{i_1<\cdots <i_k}{f}_{i_1\cdots i_k}\mathrm{sgn}(I,J){\theta }_{j_1}\wedge \cdots \wedge {\theta }_{j_{n-k}},$$here $j_1<j_2\cdots <j_{n-k}$ is the rearrangement of the complement of $i_1<\cdots <i_k$, and $\mathrm{sgn}(I,J)$ is the sign of the permutation $i_1,\cdots ,i_k,j_1,\cdots ,j_{n-k}$.

Example 1.4. We have $\ast 1\in {\mathcal{A}}^n(M)$, which is called the volume form or volume element and which will be denoted by $v_M$. $$v_M={\theta }_1\wedge \cdots \wedge {\theta }_n=\sqrt{\det g_{ij}}d{x}_1\wedge \cdots \wedge d{x}_n.$$  For a domain $D\subset M$, ${\int }_D{v}_M$ is the volume of $D$. In particular, if $M$ is compact, ${\int }_M{v}_M$ is called the volume of $M$.

Proposition 1.5. For any $f,g$ in $C^{\infty }(M)$ and for any $\omega$ and $\eta$ in ${\mathcal{A}}^k(M)$ we have

(i)	$\ast (f\omega +g\eta )=f\ast \omega +g\ast \eta$.
(ii)	$\ast \ast \omega =(-1{)}^{k(n-k)}\omega$.
(iii)	$\omega \wedge \ast \eta =\eta \wedge \ast \omega =⟨\omega ,\eta ⟩v_M$.
(iv)	$\ast (\omega \wedge \ast \eta )=\ast (\eta \wedge \ast \omega )=⟨\omega ,\eta ⟩$.
(v)	$⟨\ast \omega ,\ast \eta ⟩=⟨\omega ,\eta ⟩$.

Remark. Be careful about the sign in the boring verification.

In this chapter, from now on, we assume the manifold $M$ to be an oriented and closed Riemannian manifold.

Remark. The compactness is only needed in making ${\int }_M$ meaningful. But if we restrict the forms a little bit, the integral could be meaningful, too.

Defition 1.6. Recall the definition of $⟨\omega ,\eta ⟩$, which is a function on $M$. Now we use this to define an inner product of $\omega$ and $\eta$ on ${\mathcal{A}}^{k}M$: $$(\omega ,\eta )={\int }_M⟨\omega ,\eta ⟩v_M.$$

Proposition 1.7. $$(\omega ,\eta )={\int }_M\omega \wedge \ast \eta ={\int }_M\eta \wedge \ast \omega .$$$$(\ast \omega ,\ast \eta )=(\omega ,\eta )$$

Defition 1.8. We define $\delta =(-1{)}^k{\ast }^{-1}d\ast =(-1{)}^{n(k+1)+1}\ast d\ast$, which makes the following commutative.

Proposition 1.9. The following is true:

(i)	$\ast \delta =(-1{)}^{k}d\ast$
(ii)	$\delta \ast =(-1{)}^{k+1}\ast d$
(iii)	$\delta \circ \delta =0$
(iv)	$\delta$ is an adjoint operator of exterior differentiation $d$: $$(d\omega ,\eta )=(\omega ,\delta \eta ).$$Conversely, $d$ is an adjoint operator of $\delta$.

Remark. The proposition (iv) shows that why we compose $(-1{)}^k$ on the definition of $\delta$.

Defition 1.10. Laplacian and Harmonic form

  For a Riemannian manifold $M$, the operator defined by $$\Delta =d\circ \delta +\delta \circ d:{\mathcal{A}}^k(M)\to {\mathcal{A}}^k(M)$$is called the Laplacian or Laplace-Beltrami operator. A form $\omega \in {\mathcal{A}}^{\ast }(M)$ such that $\Delta \omega =0$ called a harmonic form. In particular, a function such that $\Delta f=0$ is called a harmonic function.

Remark. The Laplacian is an extension of the classical Laplace operator to the case of a general Riemannian manifold.

Proposition 1.11. The Laplacian $\Delta$ has the following properties:

(i)	$\ast \Delta =\Delta \ast$. If $\omega$ is a harmonic form, then so is $\ast \omega$.
(ii)	$\Delta$ is self-adjoint, that is, $$(\Delta \omega ,\eta )=(\omega ,\Delta \eta ),\forall \omega ,\eta \in {\mathcal{A}}^{\ast }(M).$$
(iii)	A necessary and sufficient condition for $\Delta \omega =0$ is that $d\omega =0$ and $\delta \omega =0$.

Corollary 1.12. For a connected, oriented, compact Riemannian manifold $M^n$, a harmonic function on $M$ is a constant function, and a harmonic $n-$form is a constant multiple of the volume element $v_M$.

Defition 1.13. We use ${\mathbb{H}}^k(M)$ to denote all the harmonic $k-$forms on $M$, that is, $${\mathbb{H}}^k(M)=\{\omega \in {\mathcal{A}}^k(M):\Delta \omega =0\}.$$

Proposition 1.14. Since all the harmonic forms are closed forms, then the map ${\mathbb{H}}^k(M)\to H_{DR}^k(M)$ is well-defined, moreover, is injective.

  From this, we can see that ${\mathbb{H}}^k(M)$ is of finite-dimension.

Remark. (Some thoughts) There we use the fact that $H_{DR}^k(M)$ is of finite dimension as $\mathbb{R}-$linear space, which is guaranteed by de Rham theorem: $H_{DR}^k(M)$ is isomorphic to the singular cohomology group with coefficient $\mathbb{R}$: $H^k(M,\mathbb{R})$, which is of finite dimension and we name it as Betti number.

Theorem. Hodge theorem

  An arbitrary de Rham cohomology class of an oriented compact Rieamnnian manifold can be represented by a unique harmonic form.

  In other words, the natural map ${\mathbb{H}}^k(M)\to H_{DR}^k(M)$ is an isomorphism.

Theorem. Hodge decomposition

  On an oriented compact Riemannian manifold, an arbitrary $k-$form can be uniquely writted as the sum of a harmonic form, an exact form, and a dual exact form.

  In other words, ${\mathcal{A}}^k(M)={\mathbb{H}}^k(M)\oplus d{\mathcal{A}}^{k-1}(M)\oplus \delta {\mathcal{A}}^{k+1}(M).$

Theorem. Poincaré duality theorem

  For a connected compact oriented $n-$dimensional $C^{\infty }$ manifold, we define a bilinear map $$\begin{array}{rl}{H}_{DR}^k(M)\times H_{DR}^{n-k}(M)& \to \mathbb{R}\\ ([\omega ],[\eta ])& \mapsto {\int }_M\omega \wedge \eta .\end{array}$$  The bilinear map defined above is nondegenerate and hence induces an isomorphism $$H_{DR}^{n-k}(M)\cong (H_{DR}^k(M){)}^{\ast }$$

Remark. Actually, the theorem above is just a special case of Poincaré duality theorem. In general, the compactity is not necessary, but the theorem need to be modified a little as a consequence.

Defition 1.15. Euler number or Euler characteristic or Euler-Poincaré characteristic.

  $\chi (K)=\sum _i(-1{)}^i{\beta }_i$, ${\beta }_i=\dim H_i(K;\mathbb{R}).$

  Then for an $n-$dimensional $C^{\infty }$ manifold $M$, we have $$\chi (M)=\sum _{i=0}^n(-1{)}^i{\dim }_{DR}^i(M).$$

Theorem 1.16. The euler characteristic of an odd-dimensional closed manifold is $0$.

Summary:

(1)	For a Riemannian manifold we may identify the tangent bundle and cotangent bundle by using the metric.
(2)	For an $n-$dimensional, oriented, Riemannian manifold we have a linear operator, called the Hodge star operator, that maps $k-$forms into $(n-k)-$forms.
(3)	For a Riemannian manifold, on can define a self-adjoint operator $\Delta$, called the Laplacian or Laplace-Beltrami operator, that generalizes the classical Laplace operator.
(4)	A function $f$ is called a harmonic function if $\Delta f=0$. A differential form $\omega$ is called a harmonic form if $\Delta \omega =0$.
(5)	On a connected closed Riemannian manifold a harmonic function is a constant function. Remark. Closed manifold means the manifold is compact and without boundary.
(6)	For a connected, closed Riemannian manifold, every de Rham cohomology class is represented by a unique harmonic form. This is called the Hodge theorem. Remark. From the point of view of magnitude, we may prove that within the set of all closed forms representing a de Rham cohomology class, there is one and only one differential form that has the best shape.
(7)	For an oriented $n-$dimensional closed manifold, the $k-$dimensional de Rham cohomology group and the $(n-k)-$dimensional de Rham cohomology group are dual to each other. This is the Poincaré duality theorem.
(8)	The Euler number of an odd-dimensional closed manifold is $0$.

2Riemann geometry and vector bundle review

  It’s natural to consider the set of all tangent spaces put together-this is called the tangent bundle of a manifold. A vector bundle is a generalization. Roughly speaking, a vector bundle is what we get by lining up vector spaces of a fixed dimension for all points of the manifold.

  For research in the theory of vector bundles the characteristic classes are important because thery express, in the language of cohomology, the way the bundles are curved.

  The definition of Vector bundle could be consulted in the book Morita.

  Firstly we review something in Riemann geometry.

Defition 2.1. Connection form and Curvature form

  For a local section frame field $\{s_1,\cdots ,s_n\}$, we have ${\nabla }_X{s}_i={\omega }_i^j(X)s_j$, where the ${\omega }_i^j(X)$ is decided by $X$. Note that ${\omega }_i^j(fX)=f{\omega }_i^j(X)$, we could regard ${\omega }_i^j$ as a $1-$form.

  Define $\omega =\{{\omega }_i^j\}$ as the connection form.

  Similarly, $R(X,Y)s_i={\Omega }_i^j(X,Y)s_j$ and ${\Omega }_i^j$ is alternate and functional-linear, then we regard ${\Omega }_i^j$ as a $2-$form.

  Define $\Omega =\{{\Omega }_i^j\}$ as the curvature form.

Proposition 2.2. Structure equation

$$d\omega =\omega \wedge \omega +\Omega .$$

Proposition 2.3. Transformation rules

  Given two open subsets $U_{\alpha }$ and $U_{\beta }$ in $M$ and trivializations $${\phi }_{\alpha }:{\pi }^{-1}(U_{\alpha })\cong U_{\alpha }\times {\mathbb{R}}^n,$$$${\phi }_{\beta }:{\pi }^{-1}(U_{\beta })\cong U_{\beta }\times {\mathbb{R}}^n,$$and we use $\{s_i\}$ to denote the local frame field on $U_{\alpha }$, $\{t_j\}$ for $U\beta$.

  Then we have two transformation rules: $${\omega }_{\beta }=g_{\alpha \beta }{\omega }_{\alpha }{g}_{\alpha \beta }^{-1}+d{g}_{\alpha \beta }{g}_{\alpha \beta }^{-1},$$$${\Omega }_{\beta }=g_{\alpha \beta }{\Omega }_{\alpha }{g}_{\alpha \beta }^{-1},$$where $g_{\alpha \beta }$ is the transformation function: $t_j=g_j^i{s}_i$.

Proposition 2.4. Bianchi’s identity

$$d\Omega =\omega \wedge \Omega -\Omega \wedge \omega .$$

  Notice that ${\nabla }_X{s}_i={\omega }_i^j(X)s_j$, then we can regard $\nabla s_i$ as a $1-$form on $M$ with values in $\Gamma (E)$.

  More generally, $$\begin{array}{rl}{\nabla }_{X}s=\nabla a^i{s}_i& =X(a^i)s_i+a^i{\nabla }_X{s}_i\\ & =d{a}^i(X)s_i+a^i{\omega }_i^j(X)s_j\\ & =(d{a}^j(X)+a^i{\omega }_i^j(X))s_j.\end{array}$$  Similarly, we can regard $\nabla s:\mathcal{X}(M)\to \Gamma (E)$.

Defition 2.5. Generally, we use $${\mathcal{A}}^k(M;E)=\{\mathcal{X}(M)\times \cdots \times (X)(M)\to \Gamma (E)\}$$to represent all the $k-$forms on $M$ with values in $\Gamma (E)$.

  An arbitrary element of ${\mathcal{A}}^k(M;E)$ can be written as a linear combination of elements of the form $\theta \otimes s$($\theta \in {\mathcal{A}}^k(M),s\in \Gamma (E)={\mathcal{A}}^0(M;E)$).

  For any section $s\in \Gamma (E)$, the map $$\nabla s:X\in \mathcal{X}(M)\mapsto {\nabla }_{X}s\in \Gamma (E)$$is linear as $C^{\infty }(M)$-modules, which means $\nabla s$ is an element of ${\mathcal{A}}^1(M;E)$.

Defition 2.6. Then we have the linear map naturally: $$\nabla :\Gamma (E)\to {\mathcal{A}}^1(M;E),$$with an easy verified property: $\nabla (fs)=df\otimes s+f\nabla s$.

  $R(X,Y):\Gamma (E)\to \Gamma (E)$ is linear as $C^{\infty }-$modules, then we regard $R(X,Y)$ as an element of $\Gamma (\mathrm{End}E)$.

  Thus, $R:\mathcal{X}(M)\times \mathcal{X}(M)\to \Gamma (\mathrm{End}E)$, which means $R\in {\mathcal{A}}^2(M:\mathrm{End}E)$.

Summary:

(1)	A vector bundle comes with a manifold called the base space, to each point of which there is associated a vector space of a fixed dimension in such a way that locally it appears like a direct product of the base space and the vector space.
(2)	The tangent bundle is a vector bundle that is formed by putting the tangent spaces of a $C^{\infty }$ manifold together.
(3)	A curve is called a geodesic if its acceleration vector is parallel along the curve.
(4)	To give a connection to a vector bundle $E$ is to define the derivative ${\nabla }_{X}s$ of an arbitrary section $s$ of $E$ in the direction of an arbitrary tangent vector $X$ to the base space.
(5)	The curvature is obtained by taking the covariant exterior derivative of the connection, and it measures the curving of the vector bundle.
(6)	A connection and its curvature are locally expressed by the connection form of degree $1$ and the curvature form of degree $2$ with values in $\mathfrak{gl}(n,\mathbb{R})$; they are related by the structure equation.

3Characteristic class in Tangent bundles

  Given an arbitrary connection $\nabla$ on $\pi :E\to M$, and $R$ the induced curvation.

  Try to define a global form on $M$, then we integrate it to get a global invariant.

Defition 3.1. We use $I_n$ to denote the set of all invariant polynomials on $M_n$.

Proposition 3.2. Define $t_i(X)=\mathrm{Tr}{X}^i$, where $X$ is an $n\times n$ metrix, then we have $I_n\cong \mathbb{R}[t_1,\cdots ,t_n]$.

  For any $f\in I_n$ of degree $k$, $f({\Omega }_{\alpha })$ is a $2k-$form on $U_{\alpha }$, and $f({\Omega }_{\alpha })=f({\Omega }_{\beta })$ on $U_{\alpha }\cap U_{\beta }$ by invariance of $f$. Hence we get a globally defined $2k-$form, say $f(\Omega )\in {\mathcal{A}}^{2k}(M)$.

Proposition 3.3. $f(\Omega )\in {\mathcal{A}}^{2k}(M)$ is a closed form.

  Since $f(\Omega )$ is closed, we may consider the de Rham cohomology class $[f(\Omega )]\in H_{DR}^{2k}(M)$.

Proposition 3.4. The de Rham cohomology class $[f(\Omega )]\in H_{DR}^{2k}(M)$ is independent of the choice of the connection $\nabla$.

Defition 3.5. Characteristic class

  $[f(\Omega )]$ deponds only on $E$ and not on the connection we take.

  Hence we denote it by $f(E)$, and call it the Characteristic class of $E$ corresponding to $f$.

Proposition 3.6. Naturality of the characteristic class relative to the bundle map.

  Given a bundle map $(\tilde{h},h):(E,M,\pi )\to (E^{\prime },M^{\prime },{\pi }^{\prime })$, that is $\tilde{h}:E\to E^{\prime }$, $h:M\to M^{\prime }$ and ${\pi }^{\prime }\circ \tilde{h}=h\circ \pi$.

  We have $f(E)=h^{\ast }(f(E^{\prime }))\in H_{DR}^{2k}(M).$

  In particular, for $g:N\to M$, $f(g^{\ast }E)=g^{\ast }(f(E))\in H_{DR}^{2k}(N)$.

Proposition 3.7. Induced connection.

  Follow the notation of the last proposition, given a connection $\nabla$ on $E^{\prime }$, we could define an induced connection ${\tilde{h}}^{\ast }\nabla$ on $E$.

  Note that ${\tilde{h}}_{E_p}$ is a linear isomorphism, define ${\tilde{h}}^{\ast }(s)(p)={\tilde{h}}_{E_p}^{-1}(s(h(p)))$, for any point $p$ on $M$. And any arbitrary section of $E$ can be locally expressed as a linear combination of those induced sections where coefficients are functions on $M$.

  Then we define $$({\tilde{h}}^{\ast }\nabla {)}_X({\tilde{h}}^{\ast }s)={\tilde{h}}^{\ast }({\nabla }_{h_{\ast }(X)}s).$$

Defition 3.8. Metric connection.

  We say a connection is compatible with the given Riemannian metric, and call it metric connection, if $$X<s,s^{\prime }>=<{\nabla }_{X}s,s^{\prime }>+<s,{\nabla }_X{s}^{\prime }>,$$holds for any $X\in \mathcal{X}(M)$, $s,s^{\prime }\in \Gamma (E)$.

Proposition 3.9. If $f$ has odd degree, then $[f(\Omega )]=0\in H_{DR}^{2k}(M)$.

Proposition 3.10. Both $\omega$, $\Omega$ are skew-symmetric metrix with values in ${\mathcal{A}}^{\ast }(M)$ with respect to a metric connection.

Proof.   Choose $\{s_i\}$ an orthonomal section frame field.

$$0=X<s_i,s_j>=<{\nabla }_X{s}_i,s_j>+<s_i,{\nabla }_X{s}_j>=<{\omega }_i^k(X)s_k,s_j>+<s_i,{\omega }_j^k(X)s_k>={\omega }_i^j(X)+{\omega }_j^i(X).$$$${\Omega }_i^j=d{\omega }_i^j-{\omega }_i^k\wedge {\omega }_k^j=-d{\omega }_j^i-{\omega }_k^i\wedge {\omega }_j^k=-d{\omega }_j^i+{\omega }_j^k\wedge {\omega }_k^j=-{\Omega }_j^i.$$

$\square$

Defition 3.11. $$\det (I+tX)=1+t\sigma (X)+t^2{\sigma }_2(X)+\cdots +t^n{\sigma }_n(X).$$$$p_k=\frac{1}{(2\pi {)}^{2k}}{\sigma }_{2k}.$$

Remark. I haven’t find out why there is a weird constant. The author says that if we choose this constant, the $p_k(E)$ is in $H^{4k}(M,\mathbb{Z})$, and I don’t know why.

Defition 3.12. Pontrjagin class

  $p_k(E)\in H_{DR}^{4k}(M)=H^{4k}(M;\mathbb{R})$ and is called the Pontrjagin class of degree $k$.

  $p(E)=[\det (I+\frac{1}{2\pi }\Omega )]=1+p_1(E)+\cdots +p_{[\frac{n}{2}]}(E)\in H_{DR}^{\ast }(M)$ is called the total Pontrjagin class.

  The closed form which can represent the Pontrjagin class is called Pontrjagin form.

Defition 3.13. Connection on Complex bundle

  Given a complex vector bundle $E\to M$, a connection is a connection $$\nabla :\mathcal{X}(M)\times \Gamma (E)\to \Gamma (E)$$for the underlying real vector bundle $E$ that furthermore satisfies the condition $${\nabla }_X(is)=i{\nabla }_{X}s.$$

  The additional condition is equivalent to condition ${\nabla }_X(fs)=X(f)s+f{\nabla }_{X}s$ being satisfied not only for every $f\in C^{\infty }(M)$ but also for every $f\in C^{\infty }(M;\mathbb{C})$.

  Or we can say that a connection $\nabla :\Gamma (E)\to {\mathcal{A}}^1(M;E)$ is a complex linear map such that $$\nabla (fs)=df\otimes s+f\nabla s,$$for all $f\in C^{\infty }(M:\mathbb{C})$ and for all $s\in \Gamma (E).$

Proposition 3.14. The structure equation, the transformation formulas and the Bianchi identity hold similarly.

Proposition 3.15. Similarly, we can prove that for any invariant polynomial $f\in I_n(\mathbb{C})$ of degree $k$, we have $f(\Omega )\in {\mathcal{A}}^{2k}(M;\mathbb{C})$.

(i)	It is a closed form;
(ii)	Its corresponding de Rham cohomology class $[f(\Omega )]\in H_{DR}^{2k}(M;\mathbb{C})$ is determined independently of the choice of the connection. We call it the characteristic class of $E$ corresponding to $f$, and denote it by $f(E)$.
(iii)	The characteristic class is natural with respect to bundle maps;
(iv)	The characteristic class corresponding to an invariang polynomial of odd degree may be not trivial.

Defition 3.16. Chern class

  For an $n-$dimensional complex vector bundle $\pi :E\to M$, the characteristic class corresponding to $$(\frac{-1}{2\pi i}{)}^k{\sigma }_k\in I_n(\mathbb{C})$$is written $$c_k(E)\in H^{2k}(M;\mathbb{R})$$and called the Chern class of degree $k$.

  In terms of the curvature form $\Omega$ we have $$[\det (I-\frac{1}{2\pi i}\Omega )]=1+c_1(E)+c_2(E)+\cdots +c_n(E)\in H^{\ast }(M;\mathbb{R}),$$which we call the total Chern class and denote by $c(E)$.

  A clasoed form representing each Chern class corresponding to a chosen connection is called a Chern form.

Remark. The $c_k(E)$ is in $H^{2k}(M;\mathbb{R})$ is a result of the next proposition.

Proposition 3.17. Each Chern class $c_k$ is a real cohomology class.

Proposition 3.18. Both $\omega$ and $\Omega$ are skew-Hermitian with respect to a Hermitian metric.

Defition 3.19. Whitney sum

  Given two vector bundles ${\pi }_1:E_1\to M$ and ${\pi }_2:E_2\to M$, define $E_1\otimes E_2=\{(u_1,u_2)\in E_1\times E_2\mid {\pi }_1(u_1)={\pi }_2(u_2)\}$ as the Whitney sum of $E_1$ and $E_2$, and the projection is $$\begin{array}{rl}\pi :E_1\oplus E_2& \to M\\ (u_1,u_2)& \mapsto {\pi }_1(u_1)={\pi }_2(u_2)\end{array}$$  Note that $\dim E_1\otimes E_2=\dim E_1+\dim E_2$.

Proposition 3.20. $E$ and $F$ are complex vector bundles.

  Clearly, $\Gamma (E\oplus F)=\Gamma (E)\times \Gamma (F)$. It follows that if $\nabla$ and ${\nabla }^{\prime }$ are the connections of $E$ and $F$, then there is a natural direct sum connection $\nabla \oplus {\nabla }^{\prime }$ on $E\oplus F$.

  If $\Omega$ and ${\Omega }^{\prime }$ are the curvature forms of $\nabla$ and ${\nabla }^{\prime }$, then the curvature form $\widetilde{Omega}$ of $\nabla \oplus {\nabla }^{\prime }$ is the direct sum matrix of $\Omega$E and ${\Omega }^{\prime }$: $$\tilde{\Omega }=\left(\begin{array}{cc}\Omega & 0\\ 0& {\Omega }^{\prime }\end{array}\right).$$

Theorem 3.21. If $E$ and $F$ are complex vector bundles, then $$c_k(E\oplus F)=\sum _{i=0}^k{c}_i(E)c_{k-i}(F),$$namely, $c(E\oplus F)=c(E)c(F)$.

Theorem 3.22. If $E$ and $F$ are real vector bundles, then $$p_k(E\oplus F)=\sum _{i=0}^k{p}_i(E)p_{k-i}(F),$$namely, $p(E\oplus F)=p(E)p(F)$.

Proposition 3.23. Let $E$ be a real vector bundle and $E\otimes \mathbb{C}$ its complexification. Then $$p_k(E)=(-1{)}^k{c}_{2k}(E\otimes \mathbb{C})\in H^{2k}(M;\mathbb{Z}),$$$$c_k(E\otimes \mathbb{C})=0\in H^{2k}(M;\mathbb{R}),k=1,3,5,...$$

Proposition 3.24. The conjugate bundle $\overline{E}$ of a complex bector bundle $E$ is isomorphic to the dual bundle $E^{\ast }$ of $E$.

Proposition 3.25. The Chern classes of the conjugate bundle $\overline{E}$ of a complex vector bundle are given by $$c_k(\overline{E})=(-1{)}^k{c}_k(E).$$We have also for the dual bundle $$c_k(E^{\ast })=(-1{)}^k{c}_k(E).$$

Proposition 3.26. Let $E$ be an $n-$dimensional complex vector bundle. Then, writing $p_i$ for $p_i(E)$ and $c_i$ for $c_i(E)$, we have $$\begin{array}{rl}1-p_1& +p_2-\cdots +(-1{)}^n{p}_n\\ & =(1+c_1+c_2+\cdots +c_n)(1-c_1+c_2-\cdots +(-1{)}^n{c}_n).\end{array}$$$$c(E_{\mathbb{R}}\otimes \mathbb{C})=c(E\oplus \overline{E})=c(E)c(\overline{E}).$$

Let $\pi :E\to M$ be a real oriented vector bundle.

Proposition 3.27. If we introduce a Riemannian metric in $E$ and select a metric connection, then the curvature form is represented by a skew-symmetric matrix. In particular, the Pontrjagin class of the highest degree is given by $$p_n(E)=[\det (\frac{1}{2\pi }\Omega )]=\frac{1}{(2\pi {)}^{2n}}[\det \Omega ]\in H^{4n}(M;\mathbb{R}).$$

Defition 3.28. $$Pf(X)=\frac{1}{2^{n}n!}\sum _{\sigma }\mathrm{sgn}\sigma x_{{\sigma }_2}^{{\sigma }_1}\cdots x_{{\sigma }_{2n}}^{{\sigma }_{2n-1}}.$$$Pf(X)$ is called the Pfaffian of $X$, where $X$ is a matrix. It satisfies $$Pf(T^{-1}XT)=\det T\cdot Pf(X),$$where $T\in O(2n)$. In particular, $Pf$ is invariant by $T\in SO(2n).$

  By applying the invariance property of $Pf$ to the curvature form, we can construct a $2n-$form on the whole of $M$: $$Pf(\Omega )\in {\mathcal{A}}^{2n}(M).$$and $Pf(\Omega {)}^2=\det \Omega .$

Defition 3.29. Euler form.

We can find a certain cohomology class $e(E)=H^{2n}(M,\mathbb{R})$ such that $e(E{)}^2=p_n(E)$.

  we set $eu(\Omega )=\frac{1}{(2\pi {)}^n}Pf(\Omega )$ and we have $p_n(E)=[eu(\Omega {)}^2]$. We call $eu(\Omega )$ the Euler form.

  We now define the Euler class $e(E)$ by setting $$e(E)=[eu(\Omega )]\in H^{2n}(M;\mathbb{R}).$$Then we have $p_n(E)=e(E{)}^2.$

Proposition 3.30. This definition depends neither on the choice of a Riemannian metric nor on that of a compatible connection.

Proposition 3.31. The Euler class is natural with respect to an orientation preserving bundle map.

Proposition 3.32. For an $n-$dimensional complex vector bundle $E$ we have $e(E_{\mathbb{R}})=c_n(E)$.

Proposition 3.33. Let $E$ and $F$ be two oriented vector bundles over a $C^{\infty }$ manifold $M$ with dimensions $2m$ and $2n$, respectively. Then the Whitney wum $E\oplus F$ is an oriented vector bundle of dimension $2(m+n)$, and its Euler class is given by $$e(E\oplus F)=e(E)e(F)\in H^{2(m+n)}(M;\mathbb{R}).$$

Theorem. The Gauss-Bonnet theorem.

Let $M$ be an oriented $2n-$dimensional closed $C^{\infty }$ manifold. Then we have $$(e(TM),[M])=\chi (M).$$Hence for the curvature form $\Omega$ of a connection compatible with any Riemannian metric in $TM$(in particular, for the Levi-Civita connection on $TM$ for a Riemannian manifold $M$), we have $${\int }_{M}eu(\Omega )=\chi (M).$$

Remark. The definition of Euler number: definition 1.15.

Lemma 3.34. Let $\pi :E\to M$ be an oriented, $2n-$dimensional vector bundle. If there exists a section $s\in \Gamma (e)$ that is never $0$, then $e(E)=0$.

Summary:

(1)	Characteristic classes are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The characteristic classes provide a simple test.
(2)	A characteristic class of a vector bundle associates to the bundle a cohomology class of the base space satisfying the naturality condition relative to any bundle map.
(3)	Substituting the curvature form into an invariant polynomial of degree $k$, we get a closed form of degree $2k$ on the base space; its cohomology class does not depend on the choice of a connection, and is called a characteristic class.
(4)	The Pontrjagin class is a characteristic class of a real vector bundle, and the Chern class is a characteristic class of a complex vector bundle. An even-dimensional, oriented, real vector bundle also has an Euler class as its characteristic class.
(5)	On a closed $C^{\infty }$ manifold, characteristic numbers are obtained by integrating various polynomials for characteristic classes.
(6)	For an oriented, even-dimensional compact Riemannian manifold, the integral of the Euler form is equal to the Euler number of the manifold. This is called the Gauss-Bonnet theorem.

4Fiber bundles and Characteristic classes

Defition 4.1. Product bundle

  Let $F$ be a $C^{\infty }$ manifold. A very simple example of copies of $F$ attached to all points of another manifold $B$ is the product $B\times F$. If we denote by $\pi :B\times F\to B$ the natural projection, then for each $b\in B$ we have ${\pi }^{-1}(b)=F$. We call this structure a product bundle.

Defition 4.2. Differentiable fiber bundle

  Let $F$ be a $C^{\infty }$ manifold. Suppose there are given $C^{\infty }$ manifolds $E$ and $B$ and a $C^{\infty }$ map $\pi :E\to B$. We call $\xi =(E,\pi ,B,F)$ a differential fiber bundle (or a differential $F$ bundle) if is satisfies the fowwling condition:

  (Local triviality)) For each point $b\in B$ there are an open neighborhood $U$ and a diffeomorphism $\phi :{\pi }^{-1}(U)\cong U\times F$ such that for an arbitrary $u\in {\pi }^{-1}(U)$ we have $\pi (u)={\pi }_1\circ \phi (u)$, where ${\pi }_1:U\times F\to U$ denots the projection onto the first component.

Remark. If we just assume that $E,B,F$ are topological spaces and $\pi$ is a continuous map and $\phi$ a homeomorphism, then we obtain the definition of a fiber bundle in general.

  But in this note, we just deal with the differential fiber bundle.

Defition 4.3. Bundle map

  Let ${\xi }_i=(E_i,{\pi }_i,B_i,F)(i=1,2)$ be two fiber bundles with the same fiber. By a bundle map from ${\psi }_1$ to ${\psi }_2$, we mean $C^{\infty }$ maps $\tilde{f}:E_1\to E_2$, $f:B_1\to B_2$ such that:

  (1) ${\pi }_2\circ \tilde{f}=f\circ {\pi }_1$;

  (2) the restriction of $\tilde{f}$ to an arbitrary fiber ${\pi }^{-1}(b),b\in B_1$, is a diffeomorphism.

  If furthermore, $f$ is a diffeomorphism, so is $\tilde{f}$ (and vice versa). Also $({\tilde{f}}^{-1},f^{-1})$ is a bundle map.

  If $B_1=B_2=B$, the two fiber bundles ${\xi }_i$ are said to be isomorphic if there exists a bundle map $\tilde{f}:E_1\to E_2$ together with the identity map $f:B\to B$. We write ${\xi }_1\cong {\xi }_2$.

  A bundle that is isomorphic to the product bundle $B\times F$ is called a trivial bundle.

Defition 4.4. Restricted bundle

  Let $\xi =(E,\pi ,B,f)$ be a fiber bundle. For any submanifold $M$ of the base $B$, the collection $$\xi {|}_M=({\pi }^{-1}(M)=E{|}_M,\pi {|}_{{\pi }^{-1}(M)},M,F)$$is also a fiber bundle with fiber $F$. We call $\xi {|}_M$ the restriction of $\psi$ to $M$.

  If there exists an isomorphism $E{|}_M\cong M\times F$, we say that $\psi$ is trivial over $M$.

Defition 4.5. Cross section

  A $C^{\infty }$ map $s:B\to E$ such that $\pi \circ s=\mathrm{id}$.

Remark. A fiber bundle may not have a cross section, and whether it have or not is an important question.

Defition 4.6. Transition function

  By definition, there exist an open covering $\{U_{\alpha }\}$ of the base space $B$ and a trivialization $${\phi }_{\alpha }:{\pi }^{-1}(U_{\alpha })\cong U_{\alpha }\times F.$$Then the map $${\phi }_{\alpha }\circ {\phi }_{\beta }^{-1}:(U_{\alpha }\cap U_{\beta })\times F\cong (U_{\alpha }\cap U_{\beta })\times F$$gives isomorphism of the trivial $F$ bundle over $(U_{\alpha }\cap U_{\beta })$. Assume $b\in U_{\alpha }\cap U_{\beta }$ and $p\in F$. Then ${\phi }_{\alpha }\circ {\phi }_{\beta }^{-1}(b,p)$ is of the form $(b,q)$, where $q$ can be written as $g_{\alpha \beta }(b)(p)$. This means that there exists a map $$g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to \mathrm{Diff}F$$such that $${\phi }_{\alpha }\circ {\phi }_{\beta }^{-1}(b,p)=(b,g_{\alpha \beta }(b)(p))(b\in U_{\alpha }\cap U_{\beta },p\in F).$$Here $g_{\alpha \beta }$ is differentiable in the sense that the map $$(U_{\alpha }\cap U_{\beta })\times F\ni (b,p)\mapsto g_{\alpha \beta }(b)(p)\in F$$is $C^{\infty }$. We call $g_{\alpha \beta }$ the transition functions of the fiber bundle.

Defition 4.7. cocycle condition

  The family of transition functions $g_{\alpha \beta }$ clearly satisfies $$g_{\alpha \beta }(b)g_{\beta \gamma }(b)=g_{\alpha \gamma }(b)(b\in U_{\alpha }\cap U_{\beta }\cap U_{\gamma }),$$called the cocycle condition.

Proposition 4.8. Given an open covring $\{U_{\alpha }\}$ of a $C^{\infty }$ manifold $B$ and a family of differentiable functions $\{g_{\alpha \beta }\}$ satisfying the cocycle condition, there is a fiber bundle $\xi =(E,\pi ,B,F)$ with $B$ as the base, with $F$ as the fiber, and with $g_{\alpha \beta }$ as the transition functions.

Remark. This means that an arbitrary $F$ bundle can be constructed by pasting together the product bundles $U_{\alpha }\times F$ by means of the transition functions.

  Here we may use any element of the group $\mathrm{Diff}F$, but most important are the cases where a centain Lie group $G$ acts on $F$.

  For example, $SO(n+1)\subset \mathrm{Diff}{S}^n$.

Defition 4.9. Structure

  Let $(E,\pi ,B,F)$ be a fiber bundle. Suppose $B$ admits an open covering $\{U_{\alpha }\}$ and each $U_{\alpha }$ admits a trivialization ${\phi }_{\alpha }:{\pi }^{-1}(U_{\alpha })\to U_{\alpha }\times F$.

  Furthermore, assume that the corresponding transition function $g_{\alpha \beta }\to \mathrm{Diff}F$ define $C^{\infty }$ maps of $U_{\alpha }\cap U_{\beta }$ into a Lie transformation group $G\subset \mathrm{Diff}F$, In this case, we say that $\{U_{\alpha },{\phi }_{\alpha }\}$ defines a $G$ structure in $(E,\pi ,B,F)$ with structure group $G$. We denote the fiber bundle by $(E,\pi ,B,F,G)$ in this case.

Defition 4.10. Admissible

  We say that a trivialization $\phi :{\pi }^{-1}(U)\to U\times F$ is admisible or compatible with the $G$ structure if there is a map $g_{\alpha }:U\cap U_{\alpha }\to \mathrm{Diff}F$ such that $\phi \circ {\phi }_{\alpha }^{-1}(b,p)=(b,g_{\alpha }(b)(p))(b\in U\cap U_{\alpha },p\in F)$ and such that the image of every $g_{\alpha }$ is contained in $G$ and the map $g_{\alpha }:U\cap U_{\alpha }\to G$ is of $C^{\infty }$.

Remark. Thus, we can talk about a maximal family of transition functions for a given $G$ structure.

Remark. For two fiber bundles with the same structure group, the definition of the bundle map is more strict. Namely, we need the "transition function" between the two bundle is of $C^{\infty }$ and with image in $G$.

Defition 4.11. Reducible

  If the image of all transition function $g_{\alpha \beta }$ lie in a Lie subgroup $H$ of $G$, thenw e may regard $\xi$ as a fiber bundle with structure group $H$. In this case, we say that the structure group $G$ of $\xi$ is reducible to $H$.

  In this terminology, we may state that a fiber bundle $\xi$ is trivial if and only if the structure group is reducible to the trivial subgroup.

Proposition 4.12. Given an open covring $\{U_{\alpha }\}$ of a $C^{\infty }$ manifold $B$ and a family of differentiable functions $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G$ satisfying the cocycle condition, we can construct a fiber bundle $\xi =(E,\pi ,B,F,G)$ with structure group $G$ whose transition functions are exactly the given $g_{\alpha \beta }$.

Defition 4.13. Associated bundles

  We say $\xi =(E,\pi ,B,F,G)$ and ${\xi }^{\prime }=(E,\pi ,B,F^{\prime },G)$ are mutually associated bundles.

Defition 4.14. Induced bundles

  Let $f$ be a differential map $f:M\to B$, and then $\{f^{-1}(U_{\alpha })\}$ is an open covering of $M$. The map $g_{\alpha \beta }\circ f:f^{-1}(U_{\alpha })\cap f^{-1}(U_{\beta })\to G$ is of class $C^{\infty }$ and satisfies the cocycle condition.

  Then there is a fiber bundle over $M$ with $\{g_{\alpha \beta }\}$ as transition functions. This bundle is called the induced bundle or the pull-back, and is denoted by $f^{\ast }(\xi )$.

  By definition, there is a natural bundle map $f^{\ast }(\xi )\to \xi$. A concrete description is given by noticing that the total space $f^{\ast }E$ of $f^{\ast }\xi$ can be put in the form $$f^{\ast }E=\{(p,u)\in M\times E;f(p)=\pi (u)\}.$$

Defition 4.15. Principle bundle

  Let $G$ be a Lie group. Then a fiber bundle $(P,\pi ,M,G,G)$ with fiber $G$ and structure group $G$ is called a principal bundle if the action of $G$ on itself is left translation, that is, $L_a:x\mapsto ax$, where $a,x\in G$. It is also called a principal $G$ bundle.

  We shall use $(P,\pi ,M,G),\pi :P\to M$, or simply $P$ to denote a principal bundle.

Proposition 4.16. Let $\xi =(P,\pi ,M,G)$ be a principal $G$ bundle. Then we can define an action of $G$ on the total space $P$ to the right, that is, a map $P\times G\ni (u,g)\mapsto ug\in P$ such that $(ug)h=u(gh)$, where $g,h\in G$.

  This action takes each fiber onto itself, and is free, that is, if $ug=u$ for some $u\in P$, then $g=e$. Further, the quotient space $P/G$ is identified with the base space $M$.

Remark. The definition of the right action is due to the locally trivialization ${\pi }^{-1}(U_{\alpha })\to U_{\alpha }\times G$.

Proposition 4.17. Conversely, suppose we are given a differential map $\pi :P\to M$ and right action of $G$ on $P$. Assume that for any point $p\in M$ there are an open neighbeihood $U$ and a diffeomorphism $\phi :{\pi }^{-1}(U)\cong U\times G$ satisfting $\pi (ug)=\pi (u)$, $\phi (ug)=\phi (u)g$, $u\in {\pi }^{-1}(U),g\in G$.

  Then we can make $(P,\pi ,M,G)$ into a principal $G$ bundle.

Proposition 4.18. For a principal bundle to be trivial it is necessary and sufficient that it admits a section.

Example 4.19. Let $\pi :P\to M$ be a principal $G$ bundle. The bundle induced by the projection ${\pi }^{\ast }P\to P$ is trivial. In fact, $P\ni u\mapsto (u,u)\in {\pi }^{\ast }P$ is a section.

  Given two smooth manifolds $F$ and $B$, it is a fundamentally important problem to classify all isomorphism classes of all fiber bundles with base $B$ and fiber $F$.

  It is in general an extremely difficult problem. Complete solutions for an arbitrary manifold $B$ are known only in a couple of cases including $F={\mathcal{S}}^1$, as we discuss in the following subsecton.

Defition 4.20. characteristic class

  Let $A$ be an abelian group. Suppose for an arbitrary fiber bundle $\xi =(E,\pi ,B,F)$ with fiber $F$ an element $\alpha (\xi )$ of the cohomology group $H^k(B;A)$ of the base $B$ with coefficients $A$ is defined and is natural relative to a bundle map in the following sense.

  Then $\alpha (\xi )$ is called a characteristic class of the $F$ bundle (of degree $k$ with coefficients $A$). Here naturality relative to a bundle means that for any bundle map between two $F$ bundles ${\xi }_i(E_i,{\pi }_i,B_i,F)(i=1,2)$ we have $\alpha ({\xi }_1)=f^{\ast }(\alpha ({\xi }_2))$.

Remark. By definition, any two isomorphic $F$ bundles over the same base space have the same characteristic classes.

  However, the hope of classfying fiber bundles by characteristic classes is difficult to fulfill, due to the fact that $\mathrm{Diff}F$ is essentially infinite-dimensional.

  On the other hand, if we restrict structure groups to Lie groups, we get a relatively satisfactory theory concerning the classification of fiber bundles and the construction of characteristic classes.

  In the remaining part of this chapter we use differential forms to build the theory of characteristic classes of fiber bundles whose structure groups are Lie groups.

Theorem 4.21. Classifying space of fiber bundles.

  Let $G$ be a Lie group. Then there exists a principal $G$ bundle $\pi :EG\to BG$, called a universal $\mathbf{G}$-bundle, for which the following is valid.

  Let $M$ be a smooth manifold and $\xi$ an arbitrary principal $G$ bundle over $M$. Then there is a differentiable map $f:M\to BG$, unique in the sense of homotopy, such that the pull-back by $f$ of the universal $G$ bundle is isomorphic to $\xi$.

  Hence there exists a one-to-one natural correspondence between the set of isomorphism classes of principal $G$ bundles over $M$ and the set of all homotopy classes $[M,BG]$ of the maps of $M$ into $BG$.

  The space $BG$ above is called the classifying space.

  The characteristic classes of a principal $G$ bundle are nothing but elements of the cohomology group of the classifying space $BG$.

Example 4.22. Tangent frame bundle

  Denote by $F_P$ the set of all frames at $p$. We also consider the space $F(M)$ of all frames at all points of $M$.

  The natural projection $\pi :F(M)\to M$ is defined by $\pi (u)=p$ if $u\in F_p$. It is easy to check that $F(M)$ is naturally a smooth manifold as in the case of $TM$. That is, if $(U,x_1,\cdots ,x_n)$ is a local coordinate system aand if for a frame $u=[v_1,\cdots ,v_n]$ we write each $v_i\in T_{p}M$ in the form $$v_i=\sum _j{g}_{ji}\frac{\partial }{\partial x_j},$$and define the map $${\pi }^{-1}(U)\to U\times GL(n;\mathbb{R})$$by $u\in F_p\to (p,g_{ij}))\in U\times GL(n;\mathbb{R})$, then it is obviously bijective.

  Now it is easy to check that $\pi :F(M)\to M$ is a principal bundle with structure group $GL(n;\mathbb{R})$. We call it the tangent frame bundle.

Defition 4.23. Connection

  Let $\xi =(E,\pi ,B,F)$ be a fiber bundle with $C^{\infty }$ manifold $F$ as fiber. If at each point $u\in E$ we can choose a subspace $H_u$ of $T_{u}E$ in such a way that $u\mapsto H_u$ is $C^{\infty }$ and $H_u$ is transversal to the fiber (That is, $T_{u}E=H_u\otimes V_u$ is the direct sum), then we say that $\xi$ is given a connection.

Remark. A connection in a fiber bundle is nothing but a distribution on $E$ that is transversal to every fiber.

Proposition 4.24. An arbitrary fiber bundle admits a connection.

Defition 4.25. If a connction is given, an arbitrary tangent vector $X\in T_{u}E$ is uniquely decomposed as $$X=X_h+X_v(X_h\in H_u,X_v\in V_u).$$We call $X_h$ and $X_v$ the horizontal and vertical components of $X$, respectiverly.

  Gievn $c:[a,b]\to B$ be a smooth curve in the base $B$. A lift curve $\tilde{c}:[a,b]\to E$ is called a horizontal lift if the velocity vector $\dot{\tilde{c}}(t)$ is horizontal for very $t$.

Proposition 4.26. Let $\xi =(E,\pi ,B,F)$ be a fiber bundle with a compact $C^{\infty }$ manifold $F$ as fiber, and fix a connection in $\xi$. Let $c:[a,b]\to B$ be a piecewise $C^{\infty }$ curve such that $b_0=c(a)$ and $b_1=c(b)$. Then for any point $u_0\in E_{b_0}$ there is a unique horizontal lift $\tilde{c}:[a,b]\to E$ such that $\tilde{c}(a)=u_0$.

Defition 4.27. Parallel displacement

From this proposition, we can obtain a map $h_c:E_{b_0}\to E_{b_1}$ by moving $u_0$ through $E_{b_0}$. This is called parallel displacement.

Proposition 4.28. $h_{c^{-1}}=h_c^{-1}$.

Remark. If the fiber $F$ is not compact, parallel displacement may not be defined. (The proposition failed.)

  A principal bundle has its structure group $G$ acting on the entire space to the right. It is then natural to relate a connectionn to the action of $G$. It becomes possible to define parallel displacement without assuming that $G$ is compact.

Defition 4.29. Connections in principal bundles

  Let $\xi =(P,\pi ,M,G)$ ne a principal bundle with structure group $G$. A connection on $\xi$ is a rule to assign a subspace $H_u$ of $T_{u}P$ at each point $u\in P$ in such a way that the following conditions satisfied:

(i)	$H_u$ is transversal to the fiber, that is, $T_{u}P=H_u\oplus V_u$;
(ii)	$\{H_u\}$ is invariant by the right action of $G$, that is, if $R_g:P\to P(g\in G)$ is defined by $R_g(u)=ug(u\in P)$, then $H_{ug}=(R_g{)}_{\ast }{H}_u$;
(iii)	$H_u$ is differentible in $u$.

  The only difference from the definition in the case of general fiber bundles is adding the second condition on invariance by the structure group $G$ acting on the right. But it is a strong condition that makes it possible to develop the theory of characteristic classes for principal bundle.

Proposition 4.30. Let $\xi =(P,\pi ,M,G)$ be a principal $G$ bundle. Given $c:[a,b]\to B$ a piecewise $C^{\infty }$ curve and an arbitrary point $u_0\in {\pi }^{-1}(c(a))$, there is a unique horizontal lift $\tilde{c}:[a,b]\to P$ such that $\tilde{c}(a)=u_0$. It is thus possible to define parallel displacement $h_c:P_{c(a)}\to P_{c(b)}$ along $c$.

  Thus, given a connection in a principal bundle, arbitrary smooth curve in the base space $M$ can be lifted to horizontal curves in the space $P$, by means of which parallel displacement of each point in the fiber is defined.

Defition 4.31. Basic knowledge about Lie algebra.

  Let $G$ be a Lie group. The tangent space $T_{e}G$ at the identity $e$ of $G$ is called the Lie algebra of $G$ and usually donoted by the corresponding German letter $\mathfrak{g}$.

  An arbitrart element $X\in \mathfrak{g}$ can be considered to be a vector field on $G$ by putting $X(g)=(L_g{)}_{\ast }X$. Here $(L_g{)}_{\ast }:T_{e}G\to T_{e}G$ is the differential of $L_g$ at $e$.

  The vector field $X$ obtained in this way iis left-invariant, that is $(L_g{)}_{\ast }X=X$ for an arbitrary $g\in G$. Since it is obvious that all left-invariant vector fields on $G$ are obtained in this way, $\mathfrak{g}$ can be considered as the set of all left-invariant vector fields on $G$.

  For $X,Y\in \mathfrak{g}$, their bracket $[X,Y]$ belongs to $\mathfrak{g}$, since it is also left-invariant. Equipped with this product, $\mathfrak{g}$ becomes a Lie algebra.

Proposition 4.32. $$g^{\ast }=the\; set\; of\; all\; left-invariant\; 1-forms\; onG.$$  A differntial form $\omega$ on $G$ is called a left-invariant differential form, if $L_g^{\ast }\omega =\omega$ for an arbitrary $g\in G$. It is obvious that the left-invariant differential forms are determined only by the value at the identity $e$. Now we can consider an arbitrary element $\omega$ in the dual space ${\mathfrak{g}}^{\ast }$ of the Lie algebra $\mathfrak{g}$ as a left-invariant 1-form on $G$.

  Practically, it is enough to set $\omega (X)=\omega ((L_g^{-1}{)}_{\ast }X)$ for $X\in T_{g}G$.

  It might be obvious that they exhaust all the left-invariant 1-forms on $G$. A left-invariant 1-form on $G$ is called a Maurer-Cartan form.

Proposition 4.33. Since $\omega (X)$ and $\omega (Y)$ is constant on $G$, we have $$d\omega (X,Y)=-\frac{1}{2}\omega ([X,Y]).$$

Defition 4.34. Maurer-Cartan form

  Let $\omega \in {\mathcal{A}}^1(G;\mathfrak{g})$ be a $\mathfrak{g}$-valued 1-form on $G$ such taht $\omega (A)=A$ for $A\in \mathfrak{g}$. This $\omega$ is also called a Maurer-Cartan form.

Proposition 4.35. $\omega \in {\mathcal{A}}^1(G;\mathfrak{g})$ is a Maurer-Cartan form, then we have the Maurer-Cartan equation in the following form: $$d\omega =\frac{1}{2}[\omega ,\omega ].$$Here, $[\omega ,\omega ]$ denots a $\mathfrak{g}$-valued 2-form defined by $$[\omega ,\omega ](X,Y)=[\omega (X),\omega (Y)]$$for arbitrary vector fields $X,Y$.

Summary

(1)	A fiber bundle is made up by manifolds, called fibers, on to each point of a manifold, called the base space, in such a way that locally it is a product. The way the fibers are arranged is described by a transformation group, called the structure group.
(2)	A fiber bundle is called a principal bundle if the structure group is a Lie group together with its natural action on the fibers to the left.
(3)	An oriented ${\mathcal{S}}^1$ bundle is completely classified by its Euler class.
(4)	For a vector field with a finite number of singular points on a closed $C^{\infty }$ manifold, the sum of the indices at singular points is equal to the Euler number. This is called the Hopf Index Theorem.
(5)	A connection in a principal bundle is a distribution of horizontal subspaces at each point of the total space that is invariant by the action of the structure group to the right.
(6)	The curvature is defined as the derivative of a connection. It describes the way a principal bundle is curved.
(7)	For a Lie group $G$ with Lie algebra $\mathfrak{g}$, $W(\mathfrak{g})={\Lambda }^{\ast }{\mathfrak{g}}^{\ast }\otimes S^{\ast }{\mathfrak{g}}^{\ast }$ is called the Weil algebra of $\mathfrak{g}$. $W(\mathfrak{g})$ plays the role of the de Rham complex of the total space of the principal $G$ bundle.
(8)	For a Lie group $G$ the subalgebra $I(G)$ of all basic elements of the Weil algebra is called the algebra of invariant polynomials of $G$. For any principal $G$ bundle, there is a homomorphism from $I(G)$ into the cohomology algebra of the base space. It is called the Weil homomorphism. Elements that are images of the Weil homomorphism are characteristic classes of the principal $G$ bundle.
(9)	The notion of a connection of a vector bundle and that of a connection in an associated principal $GL(n;\mathbb{R})$ bundle are equivalent.
(10)	A principal bundle with a connection is called a flat bundle if the curvature is $0$.