Let S be a surface of revolution in . Are the rotations about its axis all isometries of S?
Without loss of generality, let's assume that the axis of revolution is the z-axis, and the generating curve in the xz-plane is parametrised by a unit speed parametrised curve , where is Then by rotating this planar curve in the z-axis, we have the parametrisation
The coefficients of the first fundamental form of this parametrisation are Since we let the generating curve be parametrised by unit speed curve, does not depend on . We can express the rotations by angle about the z-axis in the standard basis of :
By composing this rotation with our parametrisation, we have
Hence the coefficients of the first fundamental form of the surface after rotation, are
Since , we see that the rotations of S about its axis by are local isometries to S.