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## Not connection on a principal bundle but the second fundamental form?

For a general $n+1$ manifold $\tilde{M}$ of constant curvature $k$, the curvature tensor has the form

$\tilde{R}(\tilde{X}, \tilde{Y})\tilde{Z} = k ( \tilde{g}(\tilde{X},\tilde{Z})\tilde{Y} - \tilde{g}(\tilde{X},\tilde{Z})\tilde{Y})$

With a hypersurface $M$ the second fundamental form $h$ shows itself in the change in covariant differentiations:

$\tilde{\nabla}_XY = \nabla_XY + h(X,Y)$

In particular, the correction looks like a connection of a principal bundle in the orthodox theory of gauge fields.  But in this case we have the second fundamental form of an embedding rather than a general connection form on a principal bundle.  It’s similar to the connection on a principal bundle because when the ambient manifold is a scaled four-sphere, the normal form circles of fixed diameter.