$R^n$中Signed Curvature计算及可微函数的曲率求解问询

Alright, let's tackle this question step by step—curvature calculations for submanifolds in $\mathbb{R}^n$ can get a bit abstract, but we can ground it in concrete formulas and clear definitions, focusing first on the hypersurface case (since that's where average curvature, signed curvature, and principal curvatures make the most sense as scalar quantities).

Key Precondition: Focus on Hypersurfaces #

First, a critical note: when we talk about average curvature, principal curvatures, or signed curvature, we’re almost always referring to hypersurfaces—$(n-1)$-dimensional submanifolds embedded in $\mathbb{R}^n$. For your function $f: \mathbb{R}^m \to \mathbb{R}^n$, this means we need $m = n-1$ (either as a parameterization of the hypersurface, or as a scalar function defining the hypersurface's graph). If $m < n-1$, we’re dealing with higher-codimension submanifolds, where curvature concepts shift to vector-valued quantities (like mean curvature vectors) rather than scalars.

1. Principal Curvatures & Average Curvature for Parameterized Hypersurfaces #

Suppose we have a smooth parameterization of the hypersurface: $f: \mathbb{R}^{n-1} \to \mathbb{R}^n$, where $f$ is $n$-times differentiable and the derivative $df$ has full rank ($n-1$) everywhere (so it’s an embedding).

Step 1: Compute the Fundamental Forms

• First Fundamental Form (Metric Tensor): This encodes the "intrinsic" distance on the hypersurface. For parameters $u = (u_1, u_2, ..., u_{n-1})$, it’s defined as:
  $$g_{ij} = \left\langle \frac{\partial f}{\partial u_i}, \frac{\partial f}{\partial u_j} \right\rangle$$
  where $\langle \cdot, \cdot \rangle$ is the standard Euclidean inner product in $\mathbb{R}^n$.

• Second Fundamental Form: This encodes the "extrinsic" bending of the hypersurface relative to $\mathbb{R}^n$. First, pick a unit normal vector field $N$ on the hypersurface (direction matters for signed curvature). Then:
  $$h_{ij} = \left\langle \frac{\partial^2 f}{\partial u_i \partial u_j}, N \right\rangle$$

Step 2: Find Principal Curvatures

Principal curvatures $\kappa_1, \kappa_2, ..., \kappa_{n-1}$ are the eigenvalues of the Weingarten map (shape operator) $W$, which describes how the normal field changes along tangent vectors. Equivalently, they solve the characteristic equation:
$$\det\left( h_{ij} - \kappa g_{ij} \right) = 0$$

Step 3: Compute Average Curvature

Average curvature $H$ is the arithmetic mean of the principal curvatures:
$$H = \frac{1}{n-1} \sum_{k=1}^{n-1} \kappa_k$$
Using the fundamental forms, this can also be written as:
$$H = \frac{1}{2(n-1)} g^{ij} h_{ij}$$
where $g^{ij}$ are the entries of the inverse matrix of $g_{ij}$.

2. Simplified Calculations for Graph Hypersurfaces #

If your hypersurface is defined as the graph of a scalar function $f: \mathbb{R}^{n-1} \to \mathbb{R}$ (i.e., $S = { (x_1, ..., x_{n-1}, f(x_1, ..., x_{n-1})) \mid x \in \mathbb{R}^{n-1} }$), we can simplify the formulas drastically:

• Unit Normal Vector: Choose a direction (this determines sign for signed curvature):
  $$N = \frac{(-\nabla f, 1)}{\sqrt{1 + |\nabla f|^2}} \quad \text{or} \quad \frac{(\nabla f, -1)}{\sqrt{1 + |\nabla f|^2}}$$
  where $\nabla f = \left( \frac{\partial f}{\partial x_1}, ..., \frac{\partial f}{\partial x_{n-1}} \right)$ is the gradient of $f$.

• Average Curvature:
  $$H = \frac{1}{(n-1)(1 + |\nabla f|²⁾{3/2}} \sum_{i,j=1}^{n-1} \left( \delta_{ij} - \frac{\partial f}{\partial x_i} \frac{\partial f}{\partial x_j} \right) \frac{\partial^2 f}{\partial x_i \partial x_j}$$
  Or, using the divergence operator for a more compact form:
  $$H = \frac{1}{n-1} \cdot \frac{\text{div}\left( \frac{\nabla f}{\sqrt{1 + |\nabla f|^2}} \right)}{\sqrt{1 + |\nabla f|^2}}$$

3. Signed Curvature: Definition & Computation #

Signed curvature depends entirely on the orientation of the hypersurface (i.e., the direction of the unit normal vector $N$). Here’s how it works across common cases:

For Plane Curves ($n=2$, $m=1$)

A parameterized curve $f: \mathbb{R} \to \mathbb{R}^2$ has signed curvature $\kappa_s$, whose sign tells you the direction of bending relative to the chosen normal. The formula is:
$$\kappa_s = \frac{\det(f', f'')}{|f'|^3}$$
where $f' = \frac{df}{dt}$, $f'' = \frac{d^2f}{dt2}$, and $\det$ is the 2x2 determinant. A positive $\kappa_s$ means the curve bends toward the chosen normal; negative means it bends away.

For General Hypersurfaces

For an $(n-1)$-dimensional hypersurface in $\mathbb{R}^n$, signed curvature typically refers to the signed principal curvatures (or signed average curvature). The sign is determined by the Weingarten map:

• If $W(v)$ points in the same direction as $v$, the principal curvature $\kappa$ is positive (the hypersurface bends away from the normal $N$).
• If $W(v)$ points opposite to $v$, $\kappa$ is negative (the hypersurface bends toward $N$).

To compute it:

1. Fix a unit normal direction $N$ (this sets the sign convention).
2. Calculate the principal curvatures as described earlier—their signs are the signed curvatures for each tangent direction.
3. The signed average curvature is just the average of these signed principal curvatures.

内容的提问来源于stack exchange，提问作者Francois Wassert