Regular Surface + Isometry
Let and be regular surfaces in . Let be an isometry, that is to say
for all , the tangent plane at , where I is the first fundamental form on the appropriate tangent plane, and is the differential map.
Let and be local parametrisations of at and at respectively.
I am trying to show that the gaussian curvature is the same at and I guess that this means I am trying to show that the first fundamental forms at and are the same when i put the appropriate basis vectors in, i.e.
, where and . I figured showing this would be sufficient because then the gaussian curvature is completely determined by the first fundamental form (the notation I've used means you seperate the two columns of the jacobian and use them as basis vectors).
So if I try to show this, then I end up with
and then I am totally stuck. Can anyone offer any advice? I understand this question is a bit long winded.