Hi guys, I was wondering if anyone could post or point me to a proof of the statement that given a hypersurface [itex]\Sigma [/itex], specified by setting a function [itex]f(x) = const.[/itex], the vector field [itex]\xi ^{\mu } = \triangledown ^{\mu }f = g^{\mu \nu }\triangledown _{\nu }f [/itex] will be normal to [itex]\Sigma [/itex] in the sense that [itex]\xi ^{\mu }[/itex] will be orthogonal to all [itex]u\in T_{p}(\Sigma )[/itex] for some [itex]p\in \Sigma [/itex]. I tried to visualize it for trivial manifolds but I really couldn't. If there isn't really a proof of any kind could someone at least make the statement more intuitive. Thanks in advance.(adsbygoogle = window.adsbygoogle || []).push({});

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# Normal vector fields

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