Theorems · Theorem · algebraic topology

IsContinuousRiemannianBundle.exists_continuous

∀ {B : Type u_1} {inst : TopologicalSpace B} {F : Type u_2} {inst_1 : NormedAddCommGroup F} {inst_2 : NormedSpace ℝ F}
  {E : B → Type u_3} {inst_3 : TopologicalSpace (Bundle.TotalSpace F E)} {inst_4 : (x : B) → NormedAddCommGroup (E x)}
  {inst_5 : (x : B) → InnerProductSpace ℝ (E x)} {inst_6 : FiberBundle F E} {inst_7 : VectorBundle ℝ F E}
  [self : IsContinuousRiemannianBundle F E],
  ∃ g, (Continuous fun x => ⟨x, g x⟩) ∧ ∀ (x : B) (v w : E x), inner ℝ v w = ((g x) v) w

There exists a bilinear form, depending continuously on the basepoint and defining the inner product in the fibers. This is expressed as an existence statement so that it is Prop-valued in terms of existing data, the inner product on the fibers and the fiber bundle structure.

Defined in

Mathlib.Topology.VectorBundle.Riemannian

Cited by

3 results in Mathlib

Foundations

Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

IsContinuousRiemannianBundle