Given some metric, what is an example where the length of a vector is not preserved?
if you define a metric g' = k g where k is some scalar function. you can expresss the lengths of a' = g' a_i a_j = k g a_i a_jYour question is rather ill defined. Preserved when exactly?
It is also not clear what you would mean by a ”metric having a scalar”. A metric is a type (0,2) tensor, not a scalar.