From the book of Nakahara "Geometry, Topology and Physics":(adsbygoogle = window.adsbygoogle || []).push({});

A diffeomorfism ##f:\mathcal{M}\to \mathcal{M}## is an isometry if it preserves the metric:

##

f^{*}g_{f(p)}=g_{p}

##

In components this condition becomes:

##

\frac{\partial y^{\alpha}}{\partial x^{\mu}}\frac{\partial y^{\beta}}{\partial a^{\nu}}g_{\alpha\beta}(f(p))=g_{\mu\nu}(p)

##

where x and y are the coordinates of p and f (p), respectively.

I don't understand it. I know that a metric transforms under coordinate transformation ##x^{\mu}\to x'^{\mu}## (coordinates of thesamepoint p - passive transformation). But here we have two different points. It doesn't make sense. I know that this has to do with passive vs active coordinate transformation but cannot understand it.

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# Definition of isometry in components form

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