---
tag: A9514690T3
kind: definition
name: Second fundamental form
msc: [53A05]
defines: 9KHQR16P1B
uses: [6080X3FAD4, MAJ3DH7MZS, 0BJDQ2HBH1, 05HQYM0R4G]
sources:
  - url: https://en.wikipedia.org/wiki/Second_fundamental_form
---

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]], let $x \colon U \to S$ be a local parametrisation, and let $N = x_u \times x_v / \norm{x_u \times x_v}$ be the unit [[normal vector|0BJDQ2HBH1]].
The coefficients of the *second fundamental form* of $S$ with respect to $x$ are
$$e = \ip{x_{uu}}{N}, \qquad f = \ip{x_{uv}}{N}, \qquad g = \ip{x_{vv}}{N}.$$
At $p = x(q)$, the second fundamental form is the quadratic form $I\!I_p$ on the tangent plane $T_p S$ with $I\!I_p(a x_u + b x_v) = e a^2 + 2 f ab + g b^2$.

## Remarks

One writes $I\!I = e \, du^2 + 2 f \, du \, dv + g \, dv^2$.
Equivalently, $I\!I_p(w) = -\ip{dN_p(w)}{w}$, where $dN_p$ is the differential of the unit normal field; this shows that $I\!I_p$ depends only on $S$ and the choice of $N$.
Replacing $N$ by $-N$ changes the sign of $I\!I$.
Example: for the cylinder $x(\theta, z) = (r \cos \theta, r \sin \theta, z)$ with the inner normal $N = -(\cos \theta, \sin \theta, 0)$, one gets $e = r$, $f = 0$ and $g = 0$.
For the plane, $x_{uu}$, $x_{uv}$ and $x_{vv}$ vanish for a linear parametrisation, so $I\!I = 0$.
Non-example: on a Möbius strip there is no continuous unit normal on the whole surface, so the second fundamental form is defined only locally, up to sign.
For $w \in T_p S$ with $I_p(w) = 1$, the number $I\!I_p(w)$ is the normal curvature of $S$ in the direction $w$, where $I_p$ is the [[first fundamental form|MAJ3DH7MZS]].