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.

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.)

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.

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.