Theorems · Theorem · global analysis

ContMDiffAt.inner_bundle

∀ {EB : Type u_1} [inst : NormedAddCommGroup EB] [inst_1 : NormedSpace ℝ EB] {HB : Type u_2}
  [inst_2 : TopologicalSpace HB] {IB : ModelWithCorners ℝ EB HB} {n : WithTop ℕ∞} {B : Type u_3}
  [inst_3 : TopologicalSpace B] [inst_4 : ChartedSpace HB B] {F : Type u_4} [inst_5 : NormedAddCommGroup F]
  [inst_6 : NormedSpace ℝ F] {E : B → Type u_5} [inst_7 : TopologicalSpace (Bundle.TotalSpace F E)]
  [inst_8 : (x : B) → NormedAddCommGroup (E x)] [inst_9 : (x : B) → InnerProductSpace ℝ (E x)]
  [inst_10 : FiberBundle F E] [inst_11 : VectorBundle ℝ F E] {EM : Type u_6} [inst_12 : NormedAddCommGroup EM]
  [inst_13 : NormedSpace ℝ EM] {HM : Type u_7} [inst_14 : TopologicalSpace HM] {IM : ModelWithCorners ℝ EM HM}
  {M : Type u_8} [inst_15 : TopologicalSpace M] [inst_16 : ChartedSpace HM M] [h : IsContMDiffRiemannianBundle IB n F E]
  {b : M → B} {v w : (x : M) → E (b x)} {x : M},
  ContMDiffAt IM (IB.prod (modelWithCornersSelf ℝ F)) n (fun m => ⟨b m, v m⟩) x →
    ContMDiffAt IM (IB.prod (modelWithCornersSelf ℝ F)) n (fun m => ⟨b m, w m⟩) x →
      ContMDiffAt IM (modelWithCornersSelf ℝ ℝ) n (fun b_1 => inner ℝ (v b_1) (w b_1)) x

Given two smooth maps into the same fibers of a Riemannian bundle, their scalar product is smooth.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Riemannian

Cited by

1 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceNormedAddCommGroupInnerProductSpaceFiberBundleVectorBundleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsContMDiffRiemannianBundle