- #1
synoe
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Let [itex]f:p\mapsto f(p)[/itex] be a diffeomorphism on a [itex]m[/itex] dimensional manifold [itex](M,g)[/itex]. In general this map doesn't preserve the length of a vector unless [itex]f[/itex] is the isometry.
[tex]
g_p(V,V)\ne g_{f(p)}(f_\ast V,f_\ast V).
[/tex]
Here, [itex]f_\ast:T_pM\to T_{f(p)}M[/itex] is the induced map.
In spite of this fact why [itex]ds^2=g_{\mu\nu}dx^\mu dx^\nu[/itex], called the invariance line element, doesn't change ?
[tex]
g_p(V,V)\ne g_{f(p)}(f_\ast V,f_\ast V).
[/tex]
Here, [itex]f_\ast:T_pM\to T_{f(p)}M[/itex] is the induced map.
In spite of this fact why [itex]ds^2=g_{\mu\nu}dx^\mu dx^\nu[/itex], called the invariance line element, doesn't change ?