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This is a well-known formula for hypersurfaces for which we are particularly interested on curvature $h^2$ four-sphere.  We assume the submanifold $M$ is compact of a space form of curvature $k$, with normal $n$ and a variation vector field $Y = fn$ for a function $f$ on $M$.  The second fundamental form $A(t)$ will satisfy:
$d/dt A(t) |_{t=0} = D^2 f + k f I + f A^2$
An interesting case is when $f = 1$ which provides the ‘geodesic’ normal movement where the first term vanishes and we can see the contribution of $k I$ directly.  From  my point of view, electromagnetism is not a $U(1)$ gauge theory and what we should expect is that the $U(1)$ gauge invariance is an approximation to invariance of the electromagnetic Lagrangian to these sorts of normal movements taking into account the fact that the normals individually are circles of fixed length.
We note also that if we diagonalize $A(t)$ and set $f=1$ then for $k>0$ we have that the diagonal components of $A(t)$ satisfy $a'(t) > k$ and thus the eigenvalues increase by these normal values.