The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher codimension objects, if so how is this defined? If not, how come this notion does not make much sense in higher codimensions? Any reference or explanation is appreciated, thank you.
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5$\begingroup$ It is defined as the trace of the second fundamental form, but the second fundamental form is normal bundle valued. And usually you scale by the inverse of the dimension of the submanifold. $\endgroup$– SebastianCommented Jul 16, 2020 at 6:52
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$\begingroup$ Thank you. do you happen to have reference by any chance? $\endgroup$– Johnny T.Commented Jul 16, 2020 at 7:15
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3$\begingroup$ Chapter 5.C in the Riemannian geometry book of Gallot-Hulin-Lafontaine. $\endgroup$– SebastianCommented Jul 16, 2020 at 8:46
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