is the standard orientation of the sphere inward-pointing?

Question

I realized today that I hold three inconsistent beliefs about the geometry of the "standard round sphere":

1. Its normal is outward-pointing;
2. Its mean curvature (trace of second fundamental form) is positive;
3. Its second fundamental form is the negative differential of the normal vector in the tangent directions, $II = -dr^Tdn.$

Am I wrong about one of these conventions?

Answer 1

Seems like the crux of the issue is how you define normal curvature. You can define it as $\pm II/I$. In this book, the authors couldn't decide, and provided two versions of all the important formulae.

My personal choice (and the most common one, I believe) is to define normal curvature as $II/I$, where $II = d\mathbf{X}^T\mathbf{dN}$. For the most common parameterisations $(u,v) \mapsto \mathbf{S}(u,v)$ of a sphere, the normal direction $\mathbf{N} = \mathbf{S}^u \times \mathbf{S}^v$ points outwards, away from the center. But then this means that normal curvature and mean curvature are negative everywhere. This is somewhat disturbing, but I've learned to live with it.