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- Tags: - #geomtop/differential-geometry - Refs: - #todo/add-references - Links: - ASD - flat connection - Milnor fiber - Algebraic de Rham - Local systems biject with monodromy representations
connection
Motivations
- Provides lifts of curves in \(M\) to curves in \(\mathop{\mathrm{Frame}}(M)\).
- Connects nearby tangent spaces, tangent vector fields can be differentiated as if they were functions \(f \in C^\infty(M; V)\) for a fixed vector space \(V\),
-
The main invariants of an affine connection are its curvature.
- If both vanish, \(\Gamma(TM)\) is almost a Lie algebra.
- Can define a covariant derivative
- Sometimes flat connections are referred to as integrable connections.
Definition
Horizontal sections and integrable connections
Defined as
\begin{align*}
\nabla: \Gamma({\mathbf{T}}M { {}^{ \scriptscriptstyle\otimes_{k}^{2} } }) &\to \Gamma({\mathbf{T}}M) \\ (X, Y) &\mapsto \nabla_{X} Y
,\end{align*}
where \(\Gamma\) denotes taking smooth global sections, such that for all \(f\in C^\infty(M; {\mathbf{R}})\)
- \(C^\infty(M; {\mathbf{R}})\) linear in the first variable: \begin{align*} \nabla_{f \mathrm{X}} \mathrm{Y}=f \nabla_{\mathrm{X}} \mathrm{Y} \end{align*}
- Leibniz rule in the second variable: \begin{align*} \nabla_{\mathrm{X}}(f \mathrm{Y})=\partial_{X} f \mathrm{Y}+f \nabla_{\mathrm{X}} \mathrm{Y} \end{align*} where \(\partial_X\) is the directional derivative.
Alternatively, a principal \(\operatorname{GL}_n({\mathbf{R}})\) connection on the frame bundle \(\mathop{\mathrm{Frame}}(M)\)
The Levi-Cevita connection: the unique affine torsion-free connection for which parallel transport is an isometry.
Connection-valued forms
Structure Equation
Similar to Maurer-Cartan? 
As distributions
Integrable connections
# Unsorted
- covariant derivative with respect to two variables do not need to commute. The failure of the commutativity of partial covariant derivatives is closely related to the notion of curvature.
Ricci curvature :
# Residues
Semistability, involves SNC divisors: 
Meromorphic connection
