Theorems · Theorem · geometry

Orientation.kahler_map_complex

∀ {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)) (f : E ≃ₗᵢ[ℝ] ℂ),
  (Orientation.map (Fin 2) f.toLinearEquiv) o = Complex.orientation →
    ∀ (x y : E), (o.kahler x) y = f y * (starRingEnd ℂ) (f x)

The Kahler form on an oriented real inner product space of dimension 2 can be evaluated in terms of a complex-number representation of the space.

Defined in

Mathlib.Analysis.InnerProductSpace.TwoDim

Cited by

0 results in Mathlib

Foundations

Depth 254 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFact