Theorems · Theorem · geometry

Orientation.areaForm_comp_linearIsometryEquiv

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : Fact (Module.finrank ℝ E = 2)]
  (o : Orientation ℝ E (Fin 2)) (φ : E ≃ₗᵢ[ℝ] E),
  0 < LinearMap.det ↑φ.toLinearEquiv → ∀ (x y : E), (o.areaForm (φ x)) (φ y) = (o.areaForm x) y

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

Defined in

Mathlib.Analysis.InnerProductSpace.TwoDim

Cited by

1 results in Mathlib

Foundations

Depth 253 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFact