instIsContMDiffRiemannianBundleTrivial

English

Differentiability of a map in coordinates transfers through pullback of bundles with differentiable coordinate changes.

Русский

Дифференциация в координатах сохраняется при переходе черезcoordChanges в пуллбеке расслоений.

LaTeX

$$$ContMDiffWithinAt\\;IM\\; (IB.prod 𝓘(\\mathbb{R},F)) (f)\\; s\\ x_0 \\Rightarrow ContMDiffWithinAt\\;IM\\; (IB_2.prod 𝓘(\\mathbb{R},F)) (\\lambda m, (\\varphi m)(v m))\\; s\\ x_0$$$

Lean4

/-- A trivial vector bundle, in which the model fiber has a scalar product,
is a Riemannian bundle. -/
instance : IsContMDiffRiemannianBundle IB n F₁ (Bundle.Trivial B F₁) :=
  by
  refine ⟨fun x ↦ innerSL ℝ, fun x ↦ ?_, fun x v w ↦ rfl⟩
  simp only [contMDiffAt_section]
  convert contMDiffAt_const (c := innerSL ℝ)
  ext v w
  simp [hom_trivializationAt_apply, inCoordinates, Trivialization.linearMapAt_apply]

Похожие утверждения