# 2025-01-13-10-24-58

## 2.1: Manifolds and Vector Bundles

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The most important manifolds: $C^k, C^\infty$, complex.
The most important vector bundles: $\T_X, \T_X\dual = \Omega_X, \Omega_X^k = \wedge^k \Omega_X$.
A manifold is a topological space with a set of charts $X = \Union_i U_i$ where $\phi_i: U_i \to B(0, r) = \ts{x\in \RR^n \st \abs{x} < r} \subseteq \RR^n$.
On $U_i \intersect U_j$, there are transition functions $\phi_{ij}$ which are required to be in $C^0, C^k, C^\infty$, etc.
An example: $S^1 = \RR/\ZZ$ is covered by two charts with transition functions $\phi_{ij}(x) = ax+b$.
Another example: tori $T^n = \RR^n/\ZZ^n$ where $\ZZ^n \subseteq \RR^n$ is a lattice.
Other examples include Riemann surfaces $\Sigma_g$, $\RP^2$, and the Klein bottle.
Note that a complex manifold of (complex) dimension $n$ is an orientable real manifold of (real) dimension $2n$.
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A vector bundle is a manifold $Y\mapsvia{\pi}X$ which is locally of the form $U_i \times \RR^r \to U_i$ where $r$ is the rank.
On $U_{ij}$, the transition functions are $r\times r$ matrices whose entries are functions on $U_{ij}$, i.e. $\phi_{ij}\in \GL(\OO)$ where $\OO = C^0, C^k, C^\infty$, etc.
These must satisfy a cocycle condition $\phi_{ki}\circ \phi_{jk} \circ \phi_{ij} = \id_{U_{ijk}}$.
A section is a map $s: X\to Y$ such that $\pi\circ s = \id_X$.
An example is the zero section $s(x) = 0 \in \RR^n$.
Topological vector bundles have many sections due to partitions of unity: a collection of functions $f_i$ on $X$ whose support is contained in a closed set contained in each $U_i$ with $\sum f_i = 1$.
Sections can be constructed as functions $g = (g_1, \cdots, g_r)$ which can then be multiplied by $f_i$ to extend by zero.
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Sections of $\Omega_X$ are differential forms, and sections of $\T_X$ are vector fields.
Note that $\rank \T_X = \rank \Omega_X = \dim X$.
A section of $\Omega_X$ is of the form $\omega = \sum_{i=1}^n f_i(x_1,\cdots, x_n)dx_i$.
Given transition functions $y_i = \phi_i(x_1,\cdots, x_n)$ for $1\leq i\leq n$ for charts on $X$, we have $dy_i = \sum {\partial y_i \over\partial x_j} dx_j$.
The change-of-basis matrix is the Jacobian ${\partial \phi_i \over \partial x_j}$.
A basis for $\T_X$ is ${\partial\over \partial x_i}$ for $1\leq i\leq n$.
We have ${\partial f\over \partial y_i} = \sum {\partial f\over \partial x_i} {\partial x_i\over \partial y_j}$, yielding a matrix ${\partial \phi_i\inv\over \partial y_j}$.
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Consider $X = \Sigma_g$ a Riemann surface.
Note that $H^0(\Omega_X) = \CC^g$.
If $g=0$, one has $X = S^2 = \CP^1$.
If $g=1$ then $X = T^2 = \CC/\ZZ^2$.
In the $g=1$ case, one has $H^0(\Omega_X) = \gens{\dz}_\CC$.
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