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First variation of area formula : ウィキペディア英語版
First variation of area formula
In Riemannian geometry, the first variation of area formula relates the mean curvature of a hypersurface to the rate of change of its area as it evolves in the outward normal direction.
Let \Sigma(t) be a smooth family of oriented hypersurfaces in a Riemannian manifold ''M'' such that the velocity of each point is given by the outward unit normal at that point. The first variation of area formula is
:\frac\, dA = H \,dA,
where ''dA'' is the area form on \Sigma(t) induced by the metric of ''M'', and ''H'' is the mean curvature of \Sigma(t). The normal vector is parallel to D_ \vec_ where \vec_ is the tangent vector. The mean curvature is parallel to the normal vector.
==References==

*Chow, Lu, and Ni, "Hamilton's Ricci Flow." AMS Science Press, GSM volume 77, 2006.

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