MathDB

Let

M

be a connected, compact

C^{\infty}

-differentiable manifold, and denote the vector space of smooth real functions on

M

C^{\infty}(M)

. Let the subspace

V\le C^{\infty}(M)

be invariant under

C^{\infty}

-diffeomorphisms of

M

, that is, let

f\circ h\in V

for every

f\in V

and for every

C^{\infty}

-diffeomorphism

h\colon M\rightarrow M

. Prove that if

V

is different from the subspaces

\{ 0\}

and

C^{\infty}(M)

then

V

only contains the constant functions.

Problem Statement