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+ | $$ \gdef\vect{\text{Vect}} \gdef\lcal{\mathcal L} \gdef\End{\text{End}} \gdef\Hom{\text{Hom}}$$ | ||
+ | |||
+ | |||
====== 2020-04-01, Wednesday ====== | ====== 2020-04-01, Wednesday ====== | ||
Today and Friday, we will follow Nicolascu' | Today and Friday, we will follow Nicolascu' | ||
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==== Connection and Covariant Derivative ==== | ==== Connection and Covariant Derivative ==== | ||
- | $$ \gdef\vect{\text{Vect}} \gdef\lcal{\mathcal L} $$ | + | |
Let $\pi: E \to M$ be a vector bundle. A connection should satisfies the following | Let $\pi: E \to M$ be a vector bundle. A connection should satisfies the following | ||
- Data: $\nabla: \vect(M) \times C^\infty(M, E) \to C^\infty(M, E)$, $(X, \sigma) \mapsto \nabla_X(\sigma)$. | - Data: $\nabla: \vect(M) \times C^\infty(M, E) \to C^\infty(M, E)$, $(X, \sigma) \mapsto \nabla_X(\sigma)$. | ||
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where $T^*M \ot E$ is the tensor product of two vector bundles, and $C^\infty(M, | where $T^*M \ot E$ is the tensor product of two vector bundles, and $C^\infty(M, | ||
$$\nabla: \Omega^0(M, E) \to \Omega^1(M, E). $$ | $$\nabla: \Omega^0(M, E) \to \Omega^1(M, E). $$ | ||
+ | In general, we define | ||
+ | $$ \Omega^k(M, E) = C^\infty( \wedge^k(T^*M) \ot E) $$ | ||
+ | as $k$-forms with coefficient in $E$. | ||
+ | |||
+ | ** Example 1 **: Trivial vector bundle with trivial connection. Suppose $E = M \times \R^k$, then we can define the trivial connection $\nabla$ on $E$ as | ||
+ | $$ \nabla (f_1, \cdots, f_k) = (df_1, \cdots, df_k). $$ | ||
+ | |||
+ | |||
+ | The space of connection is not a linear space, but rather, an affine linear space. Suppose $\nabla^0, \nabla^1$ are both connections on $E$. Then, for any smooth function $f \in C^\infty(M)$, | ||
+ | $$ f\nabla^0 + (1-f) \nabla^1: \Omega^0(M, E) \to \Omega^1(M, E). $$ | ||
+ | is still a connection. Indeed, the Leibniz rule works. | ||
+ | |||
+ | ** Proposition ** Suppose $\nabla^0, \nabla^1$ are both connections on $E$, then $A = \nabla^1 - \nabla^0 \in \Omega^1(M, \End(E))$. | ||
+ | |||
+ | Remark: In plain words, for any $X \in \vect(M)$, and $\sigma$ section of $E$, we have | ||
+ | $A_X(\sigma)(p)$ only depends on $\sigma(p)$, | ||
+ | |||
+ | ** Proof: ** We have $A: \Omega^0(M, E) \to \Omega^1(M, E)$ an $\R$-linear map automatically. We need to show that $A$ is $\C^\infty(M)$ linear, that is, for any $f \in C^\infty(M)$, | ||
+ | $$ A (f \sigma) = f A(\sigma). $$ | ||
+ | Indeed, this is easy to check | ||
+ | $$ A(f \sigma) = (\nabla^1 - \nabla^0) (f \sigma) = (df - df ) \otimes \sigma + f (\nabla^1 - \nabla^0) (\sigma) = f A (\sigma).$$ | ||
+ | |||
+ | ** Prop **: the space of all possible connections on $E$, denoted as $\acal(E)$ is an affine vector space model on $\Omega^1(\End(E))$. | ||
+ | |||
+ | Given the previous propostion, we only need to prove that | ||
+ | - this space $\acal(E)$ is not empty, and | ||
+ | - for any $\nabla \in \acal(E)$, $A \in \Omega^1(\End(E))$, | ||
+ | |||
+ | The first statement can be shown using partition of unity. The second statement is an easy check. | ||
+ | |||
+ | ==== Tensor, Hom and Dual ==== | ||
+ | Let $E_1, E_2$ be two vector bundles on $M$. Recall that we can define the following | ||
+ | $$ E_1 \ot E_2, \quad \Hom(E_1, E_2) $$ | ||
+ | as vector bundles on $M$, where the fiber satisfies $(E_1 \ot E_2)_p = (E_1)_p \ot (E_2)_p$ and $\Hom(E_1, E_2)_p = \Hom( (E_1)_p, (E_2)_p)$. We define the dual bundle of a vector bundle $E$ as $E^\vee := \Hom(E, \underline{\R})$ where | ||
+ | $\underline{\R} = \R \times M$ is the trivial bundle. | ||
+ | |||
+ | Suppose $(E_1, \nabla^1)$ and $(E_2, \nabla^2)$ are equipped with connections, | ||
+ | $$ \nabla( \sigma_1 \ot \sigma_2) = \nabla ( \sigma_1) \ot \sigma_2 + \sigma_1 \ot \nabla (\sigma_2) $$ | ||
+ | where we omit the superscript on $\nabla$ and we use the identification | ||
+ | $$\Omega^1(M, | ||
+ | where the tensor on the RHS is over $\C^\infty(M)$. | ||
+ | |||
+ | Suppose $T \in \Hom(E_1, E_2)$, then we define | ||
+ | $(\nabla T)(\sigma_1) = \nabla (T \sigma_1) - T (\nabla \sigma_1).$ | ||
+ | |||
+ | ==== Moving Frame ==== | ||
+ | Let $M$ be a smooth manifold of dimension $n$, $E$ a rank $r$ vector bundle on $M$ over $\R$. Suppose $(U, (x_1, | ||
+ | $$ u = \sum_\alpha u^\alpha e_\alpha = u^\alpha e_\alpha,$$ | ||
+ | using Einstein summation convention. | ||
+ | And we can write | ||
+ | $$ \nabla(u) = \nabla(u^\alpha e_\alpha) = d(u^\alpha) \ot e_\alpha + u^\alpha \ot \nabla(e_\alpha). $$ | ||
+ | Hence, if we know how $\nabla$ acts on the frame $e_\alpha$, we know how it acts on any sections. We may write | ||
+ | $$ \nabla(e_\alpha) = \Gamma^{\beta}_{i \alpha} dx^i \ot e_\beta$$ | ||
+ | These coefficients $\Gamma^{\beta}_{i \alpha}$ encodes the data of the connection over $U$. | ||
+ | |||
+ | More abstractly, we can say: given a trivilization of $E|_U = U \times \R^r$, we may consider the trivial connection $d_U$ on $U \times \R^r$, then over $U$, we have | ||
+ | $$ \nabla|_U = d_U + A_U $$ | ||
+ | where $A_U \in \Omega^1(U, \End(E))$ is called the ** local ** conection 1-form. | ||