Unit 6: Normal Curvatures, Principal Curvatures & Shape Classification

1. The Normal Curvature as a Rayleigh Quotient

At any regular point $p$ on a surface $S$, the normal curvature $\kappa_n$ in the direction $(du, dv)$ is:

Since scaling $(du, dv)$ by any non-zero constant $c \ne 0$ cancels out in the numerator and denominator, $\kappa_n$ depends only on the direction angle on the unit circle in $T_p S$. Because the unit circle is compact and $\kappa_n$ is a continuous function, $\kappa_n$ must attain an absolute maximum $\kappa_1$ and an absolute minimum $\kappa_2$.

Definition 6.1 (Principal Curvatures and Principal Directions): The maximum and minimum values of the normal curvature at $p$:

are called the principal curvatures of the surface at $p$. The directions in $T_p S$ along which these extreme values are attained are called the principal directions.

2. Derivation of the Principal Curvature Equation

To find the critical points of $\kappa_n$, we use the method of Lagrange multipliers: extremize $II(du, dv) = L du^2 + 2M dudv + N dv^2$ subject to the normalization constraint $I(du, dv) = E du^2 + 2F dudv + G dv^2 = 1$. Form the Lagrangian:

Taking partial derivatives with respect to $du$ and $dv$ and setting them to zero:

In matrix notation:

For non-trivial tangent directions $(du, dv) \ne (0, 0)$, the determinant of this coefficient matrix must vanish!

Theorem 6.1 (The Characteristic Equation for Principal Curvatures): The principal curvatures $\kappa$ are the real roots of the quadratic equation:

Expanding the determinant:

3. Orthogonality of Principal Directions

Theorem 6.2 (Orthogonality of Principal Directions): If $\kappa_1 \ne \kappa_2$, the corresponding principal directions are mutually orthogonal in the tangent plane $T_p S$.

Proof: The principal curvatures $\kappa_1, \kappa_2$ are the eigenvalues of the Shape Operator $S_p$, and the principal directions are the corresponding eigenspaces. By Theorem 5.1, $S_p$ is self-adjoint with respect to the first fundamental form inner product $\langle \cdot, \cdot \rangle$. For eigenvectors $\mathbf{v}_1, \mathbf{v}_2$ with $\kappa_1 \ne \kappa_2$:

Since $\kappa_1 - \kappa_2 \ne 0$, we must have $\langle \mathbf{v}_1, \mathbf{v}_2 \rangle = 0$. $\blacksquare$

1. Definitions of Gaussian and Mean Curvature

Dividing the quadratic characteristic equation of Theorem 6.1 by $(EG - F^2)$:

Comparing this with the standard monic quadratic $\kappa^2 - 2H\kappa + K = 0$:

Definition 6.2 (Gaussian Curvature and Mean Curvature):

1. The Gaussian Curvature $K$ is the product of the principal curvatures:

1. The Mean Curvature $H$ is the arithmetic mean of the principal curvatures:

2. Solving for Principal Curvatures in Terms of $H$ and $K$

The quadratic equation $\kappa^2 - 2H\kappa + K = 0$ has discriminant:

Notice that:

Thus, the discriminant is always non-negative, confirming that the principal curvatures are always real numbers! The roots are:

where $\kappa_1 = H + \sqrt{H^2 - K}$ is the maximum curvature and $\kappa_2 = H - \sqrt{H^2 - K}$ is the minimum curvature.

3. Gaussian Curvature as the Jacobian of the Gauss Map

Theorem 6.3 (Gauss Map Area Ratio): Let $U \subset S$ be a small region around $p$, and let $\mathbf{n}(U) \subset S^2$ be its spherical image under the Gauss map. The Gaussian curvature $K(p)$ is the limit of the ratio of the spherical area to the surface area:

with the sign determined by whether the Gauss map preserves ($K > 0$) or reverses ($K < 0$) orientation.

Proof: The area element on the surface is $dA_S = \|\mathbf{r}_u \times \mathbf{r}_v\| \, du \, dv$. The area element on the unit sphere traced by $\mathbf{n}(u, v)$ is:

By the Weingarten equations (Section 5.3), $\mathbf{n}_u = -S(\mathbf{r}_u)$ and $\mathbf{n}_v = -S(\mathbf{r}_v)$. For any linear operator $S$ on the tangent plane, the cross product transforms as:

Taking norms:

Taking the oriented limit yields the theorem. $\blacksquare$

1. Classification by the Sign of Gaussian Curvature

The sign of the Gaussian curvature $K = \frac{LN - M^2}{EG - F^2}$ dictates the local shape of the surface relative to its tangent plane. Since $EG - F^2 > 0$, the sign of $K$ is determined entirely by the sign of the discriminant of the Second Fundamental Form: $LN - M^2$.

Definition 6.3 (Classification of Surface Points): Let $p \in S$ be a regular point:

1. Elliptic Point ($K > 0$):

Both principal curvatures have the same sign ($\kappa_1 \kappa_2 > 0$). $LN - M^2 > 0$. The Second Fundamental Form $II$ is definite (positive or negative). Geometry: The surface curves away from the tangent plane in the same direction along all tangent vectors; locally, the surface lies entirely on one side of its tangent plane (like an ellipsoid or sphere).

1. Hyperbolic Point ($K < 0$):

The principal curvatures have opposite signs ($\kappa_1 \kappa_2 < 0$). $LN - M^2 < 0$. The Second Fundamental Form $II$ is indefinite. Geometry: The surface is saddle-shaped; the tangent plane cuts through the surface, dividing it into two regions lying on opposite sides of the plane (like a hyperbolic paraboloid or catenoid).

1. Parabolic Point ($K = 0$, but not all second coefficients vanish):

One principal curvature is zero and the other is non-zero ($\kappa_1 \ne 0, \kappa_2 = 0$). $LN - M^2 = 0$, with $L^2 + M^2 + N^2 > 0$. Geometry: The surface bends along one direction but is flat along another (like a cylinder or cone).

1. Planar Point ($\kappa_1 = \kappa_2 = 0$):

Both principal curvatures vanish identically ($L = M = N = 0$). Geometry: Second-order curvature is completely flat in all directions (like a point on a flat plane).

2. Umbilical Points

Definition 6.4 (Umbilical Point): A point $p \in S$ is called an umbilical point (or umbilic) if the two principal curvatures are equal:

Equivalently, the Shape Operator is a scalar multiple of the identity: $S_p = \kappa I_{\text{id}}$.

At an umbilic:

• The normal curvature is identical in every direction: $\kappa_n(\mathbf{w}) = \kappa$ for all $\mathbf{w} \in T_p S$.
• Every tangent direction is a principal direction!
• If $\kappa_1 = \kappa_2 \ne 0$, $p$ is an elliptic umbilic (e.g., any point on a sphere of radius $1/\kappa$).
• If $\kappa_1 = \kappa_2 = 0$, $p$ is a planar umbilic.

1. Definition and Physical Origin of Minimal Surfaces

Definition 6.5 (Minimal Surface): A regular surface $S$ is called a minimal surface if its Mean Curvature vanishes identically at every point:

Equivalently, the principal curvatures are equal in magnitude and opposite in sign:

Physically, by the Young-Laplace equation of fluid mechanics, the pressure difference $\Delta P$ across a soap film interface with surface tension $\sigma$ is:

When the pressure is equal on both sides ($\Delta P = 0$), the soap film spontaneously adopts a configuration with $H = 0$, minimizing its total surface area subject to the fixed boundary wire frame!

2. Properties of Minimal Surfaces

1. Gaussian Curvature is Non-Positive:

Since $\kappa_2 = -\kappa_1$, $K = \kappa_1 \kappa_2 = -\kappa_1^2 \le 0$. Therefore, every point on a minimal surface is either hyperbolic ($K < 0$) or planar ($K = 0$). There are no elliptic points on a minimal surface!

2. Minimal Surface Partial Differential Equation:

For a Monge patch $z = f(x, y)$, the condition $H = 0$ translates into Lagrange's non-linear minimal surface PDE:

3. Conformal Parametrizations and Harmonic Coordinates:

If a surface is parametrized by isothermal coordinates ($E = G = \lambda^2, F = 0$), then:

Thus, in isothermal coordinates, a surface is minimal ($H = 0$) if and only if its coordinate functions are harmonic:

3. Classical Examples of Minimal Surfaces

• The Catenoid: Discovered by Euler (1744), the only minimal surface of revolution other than the plane.
• The Helicoid: Discovered by Meusnier (1776), the only ruled minimal surface other than the plane.
• Scherk's Surface: Discovered by Scherk (1834), $z = \ln(\cos x / \cos y)$.
• Enneper's Surface: An algebraic minimal surface of degree 9.

1. Parametrization of a Surface of Revolution

Let a profile curve in the $xz$-plane be parametrized by arc-length $u$:

Rotating this curve about the $z$-axis through angle $v \in [0, 2\pi)$ produces the surface of revolution:

2. First and Second Fundamental Forms

Computing the partial derivatives:

Metric coefficients:

The coordinate grid is orthogonal ($F = 0$). The unit normal is:

Second derivatives:

Taking dot products with $\mathbf{n}$:

Since $F = 0$ and $M = 0$, the coordinate curves (meridians and parallels) are lines of curvature!

3. Principal and Gaussian Curvatures

The principal curvatures along the meridians ($dv = 0$) and parallels ($du = 0$) are:

Differentiating $(f')^2 + (g')^2 = 1$ with respect to $u$:

Substituting this into $\kappa_1$:

Multiplying $\kappa_1$ and $\kappa_2$:

Theorem 6.4 (Gaussian Curvature of a Surface of Revolution): For a surface of revolution with profile curve parametrized by arc-length $u$:

4. Beltrami's Pseudosphere

Setting $K = -1/a^2$ (constant negative curvature) in $-\frac{f''}{f} = -\frac{1}{a^2}$ gives the differential equation:

The profile curve generating the pseudosphere is the tractrix:

Beltrami demonstrated in 1868 that the pseudosphere provides a concrete local model for the hyperbolic geometry of Lobachevsky and Bolyai!