= Critical-point criterion for equal conformal connections
{title2=$\widetilde\nabla_p=\nabla_p\Longleftrightarrow(d\sigma)_p=0$}
On a positive-dimensional <Riemannian manifold>, equality means equality of the two covariant derivatives at $p$ for every pair of vector fields. A <critical point> of $\sigma$ annihilates the conformal correction. Conversely, writing that correction as $A$, the identity $g(A(v,v),v)=g(v,v)d\sigma(v)$ for all tangent vectors forces $d\sigma=0$ if $A=0$. A smooth function on a nonempty compact boundaryless manifold has a critical point, so conformally related connections coincide somewhere. Compact manifolds with boundary need not have one.
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