Q. Is there a higher-dimensional version of the theorem due to Fenchel that the total curvature of a closed curve in $\mathbb{R}^3$ is $\ge 2\pi$, with equality only if the curve is planar and convex?

Here the total curvature is $\int \kappa(s) \, ds$. Fáry & Milnor subsequently proved that if the curve is knotted, then the total curvature is strictly greater than $2\pi$. (Apologies if I have misinterpreted the history.)

By "higher-dimensional" I mean a result addressing the "total curvature" (appropriately defined) of a closed, compact (hyper-)surface in $\mathbb{R}^d$, $d>3$. In other words, I am seeking a generalization of the curve to a (hyper-)surface, rather than a generalization of a 1D curve to $\mathbb{R}^d$, which I understand is addressed by Fáry/Milnor.

Milnor, John. "On total curvatures of closed space curves." Mathematica Scandinavica (1954): 289-296. (Jstor link, eudml link.)

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    $\begingroup$ See papers of Chern and Lashof on the total curvature of immersed submanifolds. $\endgroup$ – Deane Yang Apr 30 '17 at 1:44
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    $\begingroup$ @DeaneYang: Thanks! Chern, Shiing-shen, and Richard K. Lashof. "On the total curvature of immersed manifolds." American Journal of Mathematics 79.2 (1957): 306-318. $\endgroup$ – Joseph O'Rourke Apr 30 '17 at 12:13
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    $\begingroup$ You may also like this one by Kuiper and Meeks: projecteuclid.org/download/pdf_1/euclid.jdg/1214441483 Kuiper has several nice surveys on this topic. $\endgroup$ – alvarezpaiva Apr 30 '17 at 19:03

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