I faced this sentence:
"We consider the space of Riemannian metrics modulo diffeomorphism and scaling."
Can anyone explain to me what is the meaning of modulo diffeomorphism and scaling?
Thanks!
2026-05-05 10:01:11.1777975271
Meaning of modulo diffeomorphism
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"Modulo diffeomorphism" means that two metrics are called equivalent if one of them is the pullback of the other along a diffeomorphism (of the underliyng space). It is easy to see that this is indeed an equivalence relation on the class of all Riemannian metrics.
"Modulo scaling" assumes that we regard two metrics equivalent iff they are positive multiples of each other. This is again, obviously, an equivalence relation.
If two elements of a set (class) belong to the same class of equivalence they are termed equal modulo this equivalence relation.
In a fancier language people say that we consider orbits in the space of metrics under the action of the group of diffeomorphism or the group $(\Bbb R_+, \cdot)$.