Formalize shape of variation of functionals

[rafaelrojasmiliani]

Let $(M, \mathcal{O}, \mathcal{A})$ be a smooth manifold, $\gamma: I \longrightarrow M$, be covered by $\mathcal{C} \subset \mathcal{A}$. Consider the following integral

• 1 We need to define the set of integrable curves on a manifold
  • see here

$$ F: \int_\gamma \boldsymbol{\omega} $$

• 2 We need to define the variation of a curve in a manifold
  • a
  • b
  • c