Theorems · Theorem · functional analysis

Orientation.volumeForm_comp_linearIsometryEquiv

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {n : ℕ}
  [_i : Fact (Module.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) (φ : E ≃ₗᵢ[ℝ] E),
  0 < LinearMap.det ↑φ.toLinearEquiv → ∀ (x : Fin n → E), o.volumeForm (⇑φ ∘ x) = o.volumeForm x

The volume form is invariant under pullback by a positively-oriented isometric automorphism.

Defined in

Mathlib.Analysis.InnerProductSpace.Orientation

Cited by

0 results in Mathlib

Foundations

Depth 250 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFact