Theorems · Theorem · several complex variables

MDifferentiable.exists_eq_const_of_compactSpace

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {F : Type u_2} [inst_2 : NormedAddCommGroup F]
  [inst_3 : NormedSpace ℂ F] {H : Type u_3} [inst_4 : TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless]
  {M : Type u_4} [inst_6 : TopologicalSpace M] [inst_7 : ChartedSpace H M] [IsManifold I 1 M] [CompactSpace M]
  [PreconnectedSpace M] {f : M → F}, MDiff f → ∃ v, f = Function.const M v

A holomorphic function on a compact connected complex manifold is the constant function f ≡ v, for some value v.

Defined in

Mathlib.Geometry.Manifold.Complex

Cited by

0 results in Mathlib

Foundations

Depth 293 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceModelWithCorners.BoundarylessTopologicalSpaceChartedSpaceIsManifoldCompactSpacePreconnectedSpace