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I read the following paper Three-Dimensional Metrics as Deformations of a Constant Curvature Metric and discovered the following result:

Three-Dimensional Metrics as Deformations of a Constant Curvature Metric

I have three questions:

(1) Is $h$ also a conformally flat metric?

(2) Is $\boldsymbol{s}$ strictly the gradient of a scalar function, the curl of another field, or both?

(3) Am I safe to assume that the proof of this theorem is valid? I was unable to find a peer reviewed equivalent.

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Regarding (1) a three manifold is conformally flat if and only if the Cotton tensor vanishes. The Cotton tensor is defined as a linear combination of covariant derivatives of the curvature tensor, and hence, in particular, always vanishes for constant curvature metrics. So yes, $h$ is by definition conformally flat.

Regarding (2) a quick scan of the paper reveals no statement regarding whether $s$ can be taken to be either exact or co-exact, and I don't see a priori any reason why this should be true. (But I have no time to read the paper in detail.)

Regarding (3), the paper is published in General relativity and Gravitation, Volume 34, no. 2, in February 2002. So it certainly is peer-reviewed.

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  • $\begingroup$ Thank you so much. I was confused on why they multiplied a scalar function to a constant curvature metric and not call it a conformal metric. I was unsure of what a 1-form was in terms of Helmholtz's decomposition. $\endgroup$ – linuxfreebird Apr 3 '15 at 21:28

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