Here I asked the question whether the curvature deterined the metric. Since I am unfortunately completely new to Riemannian geometry, I wanted to ask, if somebody could give and explain a concrete example to me, as far as the following is concerned:
At the MO page (as cited above) I got the following answer to the question
Given a compact Riemannian manifold M, are there two metrics g1 and g2, which are not everywhere flat, such that they are not isometric to one another, but that there is a diffeomorphism which preserves the curvature?
If the answer is yes:
Can we chose M to be a compact 2-manifold?
On the positive side, if $M$ is compact of dimension $\ge 3$ and has nowhere constant sectional curvature, then combination of results of Kulkarni and Yau show that
a diffeomorphism preserving sectional curvature is necessarily an isometry.Concerning 2-dimensional counter-examples: First of all, every surface which admits an open subset where curvature is (nonzero) constant would obviously yield a counter-example. Thus, I will assume now that curvature is nowhere constant. Kulkarni refers to Kreyszig's "Introduction to Differential Geometry and Riemannian Geometry", p. 164, for a counter-example attributed to Stackel and Wangerin. You probably can get the book through interlibrary loan if you are in the US.
I looked up the example in Kreyszig's "Introduction to Differential Geometry and Riemannian Geometry", p. 164:
If we rotate the curve $x_3=\log x_1$ about the $x_3$-axis in space,
we obtain the surface of revolution $X(u_1,u_2)=(u_2\cos(u_1), u_2\sin(u_1),\log(u_2))$, $u_2>0$.
This is diffeomorphic to the helicoid $X(u_1,u_2) =(u_2\cos(u_1),u_2\sin(u_1),u_1)$.
I think, these manifolds are not compact (but I assumed compactness of the manifold in my question on MO).
I don't understand, how to manipulate this example in order to get a compact manifold.
Thank you for your help.
Best Answer
Here's another example.
First, imagine a short cylinder $S^1\times [0,1]$ and a long cylinder $S^1\times [0,10^{10}]$, but with the same radii. Smoothly cap off both ends of the cylinders in the same way using spaces homeomorphic to discs.
The resulting manifolds are both homeomorphic to $S^2$, are not isometric (since one has a much larger diameter than the other), but there is a curvature preserving diffeomorphism between them.
To see this, just convince yourself there is a diffeomorphism $f:S^1\times[0,1]\rightarrow S^1\times [0,10^{10}]$ with the property that $f$ is an isometry when restricted to $[0,\frac{1}{4}]$ and $[\frac{3}{4},1]$. This condition allows you to extend $f$ to a curvature preserving diffeo of both compact manifolds.