Example of non homogenous manifold with a finitely generated algebra of natural functions

Question

Let $(M,g)$ be a Riemannian manifold. Let $C^{\infty}_{Nat}(M,g)$ be the $\mathbb{R}$-algebra of scalar invariants of the curvature tensor and all its higher covariant derivatives. An example of a member of $C^{\infty}_{Nat}(M,g)$ is:

$$\sum_{i,j,b,k,c,d,f=1}^nR(e_i,e_j,e_j,e_k)\nabla R(e_k,e_b,e_b,e_c,e_i)\nabla R(e_c,e_d,e_d,e_f,e_f)-\sqrt{2} \sum_{i,j=1}^n R(e_i,e_j,e_j,e_i)$$

Question: If $C^{\infty}_{Nat}(M,g)$ is finitely generated as an $\mathbb{R}$-algebra, must $(M,g)$ be homogenous ? If not, can you please provide counterexamples ?

Thank you

Answer 1

No. Here's a counterexample: Let $f(r)$ satisfy the equation $f'' + 2 f^3 = 0$ with the initial conditions $f(0)=1$ and $f'(0)=0$. Then $f$ satisfies $(f')^2+f^4=1$ and is periodic with period $$ L = 2\int_{-1}^1 \frac{df}{\sqrt{1-f^4}} > 0. $$ Moreover, $f(r)=f(-r)=-f\bigl(\tfrac12L-r\bigr)$ (so $f\bigl(\pm\tfrac14 L\bigr) = 0$) and $f'\bigl(\pm\tfrac14 L\bigr)=\mp1$.

Now consider the rotationally invariant metric $g = dr^2 + f(r)^2\,d\theta^2$. This defines a smooth metric on the $2$-sphere (the poles are at $r= \pm\tfrac14 L$) with Gauss curvature $K = 2f^2$, which is not constant. Hence, this metric is not homogeneous.

It is easy to see that $|\nabla K|^2 = 16 f^2(f')^2 = 16f^2(1-f^4)$, which is a polynomial in $K$. By induction, all of the 'natural invariants' of this metric are polynomials in $K$. Thus, the ring of natural invariants of this metric is a polynomial ring generated by $K$, and hence, it is finitely generated.

(Even if one were to include $f'$ as a 'natural invariant', the ring of such invariants is still finitely generated as an $\mathbb{R}$-algebra.)

Added remark: In fact, there are lots of examples with no nontrivial symmetries. It is not hard to show that if $M^2\subset\mathbb{R}^3$ is any smooth algebraic surface, the ring of scalar invariants (i.e., scalar contractions of $K,\nabla K,\nabla^2K,\ldots$) of the induced Riemannian metric is finitely generated.