Definition of total curvature

Question

I am reading the first chapter of the book "A course in Minimal Surfaces" by Colding and Minicozzi.

My question is about the concept of total curvature of an embedded submanifold of $\mathbb{R}^n$.

Let us restrict our attention to the simple case where $\Sigma$ is a $2$-dimensional embedded submanifold of $\mathbb{R}^3$. Let $A$ be the second fundamental form. It seems that in the book the total curvature of $\Sigma$ is $|A|^2$, i.e. the squared norm of $A$.

Is this the standard definition? What does it represent? I used to use the term "total curvature" as a synonym for the Gaussian curvature, i.e. for $K = \det A $.

Any clarification would be very appreciated! Thanks!

Answer 1

Briefly, the relation between mathematical concepts and terminology is neither single-valued nor one-to-one.

In the differential geometry of surfaces, "total curvature" does generally refer to the integral of $K\, dA$.

In the theory of minimal surfaces, the $L^{2}$-norm of the second fundamental form, i.e., the integral of $$ k_{1}^{2} + k_{2}^{2} = (k_{1} + k_{2})^{2} - 2k_{1} k_{2} = 4H^{2} - 2K, $$ is a useful functional. Though one might call this the "(mean curvature) energy", authors are free to introduce convenient terminology (within reason).