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The question is as follows: Does there exist an example of a (continuous) family of metrics $g_t$ on a compact manifold such that the following properties hold?

  • All metrics $g_t$ have constant scalar curvature and this constant is the same for all $t$.

  • All these metrics lie in the same conformal class.

  • These metrics are NOT isometric.

As far as I know there are a lot of examples (including the round sphere) satisfying the first two properties but not the third.

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    $\begingroup$ What about the family $(1+t)g_0$ on $\mathbb{T}^n$ where $g_0$ is some flat metric. $\endgroup$
    – Rbega
    Commented Dec 12, 2014 at 17:18
  • $\begingroup$ I want to rule out rescalings. So you can add the condition that the volume is preserved (This is not nessecary if $\mathrm{scal}\neq0$ because fixing the scalar curvature has the same effect). $\endgroup$
    – Klaus Kröncke
    Commented Dec 14, 2014 at 12:45
  • $\begingroup$ Maybe you should check on the Yamabe problem. Basically, you are asking if there are non trivial critical levels of the Yamabe functional on conformal structures. Whereas minima of this functional are understood, I don't know if higher critical points are. $\endgroup$
    – BS.
    Commented Dec 14, 2014 at 20:48

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