[Math] Geometrical interpretation of second fundamental form


What is the geometrical interpretation of the second fundamental form?

Let $N\subset M$ be a hypersurface and locally orient it with normal field $n$. I heard from my professor that if the 2nd fundamental form is strictly positive w.r.t. $n$, then it implies geodesical convexity, i.e. if I connect two neighboring points in $N$ with a geodesic of $M$, then the geodesic curves to the direction of $n$.

For example if $M=B^2(0, 1)$, $N=S_r=\{(x,y)\in\mathbb{R}^2:x^2+y^2=r\}, r\in]0,1[$, and I choose the inner normal field for $S_r$, then any geodesic connecting two points of $S_r$ lies completely in $B(0, r)$ except the endpoints. My question is: is this true and why? I only know this article, which says that infinitesimal convexity implies local convexity, i.e. N "lies on one side" with respect to the image of sufficiently small neighborhood under $\exp$.

Best Answer

The second fundamental form is measuring how the surface patch $\sigma(U):=V$, bends with respect to the surface normal of $V$ after a small perturbation in the domain $U$ of $\sigma$. In order to do this you measure the quantity,

$$\big(\sigma(u + \Delta u, v+ \Delta v) - \sigma(u,v) \big) \cdot \textbf{N}$$

where $\textbf{N}$ is the normal field on $V$.

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You can use Taylor's theorem to approximate $\sigma(u + \Delta u, v+ \Delta v) - \sigma(u,v)$ and that is how the second fundamental form for surfaces arises.

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