Let be a Riemannian manifold of dimension . On the boundary we have the so-called outward normal vector . Let be the second fundamental form and

is the mean curvature. Let be a metric conformally related to . The transformation law for the second fundamental form reads as follows

where is the normal derivative with respect to . Multiplying both sides of this equation with gives

that is,

since

and

If we further let

that is, , we then have

since

In other words, we may write as

See also:

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Why do I have minus in the second equation (the second fundamental form changes under confromal metric)?

Comment by Han — June 8, 2018 @ 5:25

Which equation?

Comment by Ngô Quốc Anh — June 8, 2018 @ 6:00

The second one this page. When I write down the connection change under conformal metric, it ends up with ….-g_{ij}\nabla f.

Comment by Han — June 16, 2018 @ 1:36