Unit 8: Fundamental Equations of Surface Theory & Theorema Egregium

1. The Moving Gauss Trihedron

At every regular point $p$ of a surface $S = \mathbf{r}(u, v)$, the three vectors:

form a linearly independent basis of $\mathbb{R}^3$, known as Gauss's moving frame (analogous to the Frenet-Serret frame $\{\mathbf{T}, \mathbf{N}, \mathbf{B}\}$ for space curves). Any vector in $\mathbb{R}^3$, in particular the second partial derivatives $\mathbf{r}_{uu}, \mathbf{r}_{uv}, \mathbf{r}_{vv}$, can be uniquely decomposed along this basis:

The normal components are precisely the coefficients of the Second Fundamental Form:

The coefficients of the tangential components $\Gamma_{ij}^k$ are called the Christoffel symbols of the second kind.

2. Derivation of Christoffel Symbols in Terms of the Metric $E, F, G$

Because mixed partial derivatives commute for $C^2$ surfaces ($\mathbf{r}_{uv} = \mathbf{r}_{vu}$), the Christoffel symbols are symmetric in their lower indices:

To express $\Gamma_{ij}^k$ purely in terms of the metric coefficients $E, F, G$ and their partial derivatives, take the dot product of the Gauss decomposition equations with $\mathbf{r}_u$ and $\mathbf{r}_v$: For $\mathbf{r}_{uu}$:

Recall that:

Thus, we obtain the $2 \times 2$ linear system:

Multiplying by the inverse metric matrix $g^{-1} = \frac{1}{EG - F^2}\begin{pmatrix} G & -F \\ -F & E \end{pmatrix}$:

Theorem 8.1 (Explicit Formulas for Christoffel Symbols): Letting $D = EG - F^2 > 0$:

For an orthogonal coordinate system ($F = 0$):

This shows that all six Christoffel symbols $\Gamma_{ij}^k$ are completely determined by the First Fundamental Form!

1. The Integrability Condition for Surfaces

For a smooth $C^3$ surface, the third partial derivatives of the position vector must commute:

Let us substitute the Gauss frame decompositions and Weingarten equations into $\frac{\partial}{\partial v}(\mathbf{r}_{uu}) = \frac{\partial}{\partial u}(\mathbf{r}_{uv})$:

Expanding both sides using the product rule and substituting $\mathbf{r}_{uu}, \mathbf{r}_{uv}, \mathbf{r}_{vv}, \mathbf{n}_u, \mathbf{n}_v$ gives a linear combination of $\{\mathbf{r}_u, \mathbf{r}_v, \mathbf{n}\}$. Because $\{\mathbf{r}_u, \mathbf{r}_v, \mathbf{n}\}$ is a basis of $\mathbb{R}^3$, the coefficients of $\mathbf{r}_u, \mathbf{r}_v,$ and $\mathbf{n}$ must match independently on both sides!

2. The Gauss Equation and the Codazzi-Mainardi Equations

Equating the coefficients of $\mathbf{r}_v$ yields the Gauss Equation:

Equating the normal components ($\mathbf{n}$) yields the Codazzi-Mainardi Equations:

Theorem 8.2 (The Codazzi-Mainardi Equations):

3. Bonnet's Fundamental Theorem of Surface Theory

Theorem 8.3 (Bonnet's Theorem, 1867): Let $U \subseteq \mathbb{R}^2$ be a simply connected domain. Let $E, F, G$ and $L, M, N$ be smooth functions on $U$ such that:

1. $E > 0$ and $EG - F^2 > 0$ (positive definiteness of $I$).
2. The Gauss equation is satisfied.
3. The Codazzi-Mainardi equations are satisfied.

Then there exists a smooth regular surface patch $\mathbf{r}: U \to \mathbb{R}^3$ having $I$ and $II$ as its First and Second Fundamental Forms. Furthermore, this surface is unique up to a rigid Euclidean motion (rotation and translation) in $\mathbb{R}^3$!

Bonnet's theorem is the surface analogue of the Fundamental Theorem of Space Curves (Theorem 2.5), stating that $I$ and $II$ completely characterize a surface up to rigid positioning in space.

1. Gauss's Remarkable Discovery: Theorema Egregium

In 1827, Carl Friedrich Gauss proved what he appropriately named the Theorema Egregium ("Remarkable Theorem"), one of the most celebrated results in all of mathematics. By definition, Gaussian curvature is extrinsic:

It was computed via the Second Fundamental Form $II$, which relies on the surface normal $\mathbf{n}$ and the 3D spatial embedding.

Theorem 8.4 (Gauss's Theorema Egregium, 1827): The Gaussian Curvature $K$ of a regular surface in $\mathbb{R}^3$ depends only on the coefficients of the First Fundamental Form $E, F, G$ and their first and second derivatives. That is, $K$ is an intrinsic invariant of the surface!

Proof: By the Gauss compatibility equation derived in Section 8.2:

Dividing both sides by $EG - F^2$:

Since both $LN - M^2$ and $EG - F^2$ are completely expressible in terms of $E, F, G, E_u, E_v, F_u, F_v, G_u, G_v$ and their second derivatives, $K$ depends exclusively on the metric! $\blacksquare$

Profound Corollaries:

1. Invariance Under Isometry: If two surfaces are locally isometric, they have identical Gaussian curvature at corresponding points: $K_1 = K_2$.
2. Impossible Planar Map of the Earth: A flat sheet of paper has $K = 0$. A sphere of radius $R$ has $K = 1/R^2 > 0$. Since $K_{\text{sphere}} \ne K_{\text{plane}}$, it is mathematically impossible to construct a distance-preserving flat map of any portion of the Earth without distortion!

2. The Brioschi Determinant Formula for $K$

Francesco Brioschi (1852) gave a compact determinant expression for $K$:

Theorem 8.5 (Brioschi's Formula): For any regular coordinate patch with metric $E, F, G$ and $D = EG - F^2$:

For an orthogonal coordinate patch ($F = 0$), Brioschi's formula simplifies to the famous formula:

1. Geodesics: The Straightest Curves on a Surface

Definition 8.1 (Geodesic): A regular curve $C: \mathbf{r}(s)$ parametrized by arc-length $s$ on a surface $S$ is called a geodesic if its acceleration vector $\mathbf{r}''(s)$ is perpendicular to the tangent plane $T_p S$ at every point:

Equivalently, its geodesic curvature vanishes identically: $\mathbf{k}_g = \mathbf{0}$.

Geodesics represent:

1. Curves of zero intrinsic acceleration (free-particle trajectories constrained to the surface).
2. Curves that locally minimize arc-length between two nearby points on the surface.

2. The Differential Equations of Geodesics

Let a curve be parametrized by arc-length $s$: $u = u(s), v = v(s)$. The velocity vector is $\mathbf{r}'(s) = \mathbf{r}_u u' + \mathbf{r}_v v'$. The acceleration vector is:

Substituting the Gauss frame decompositions $\mathbf{r}_{ij} = \Gamma_{ij}^1 \mathbf{r}_u + \Gamma_{ij}^2 \mathbf{r}_v + b_{ij} \mathbf{n}$:

For $\mathbf{r}''(s)$ to be parallel to $\mathbf{n}$, its tangential components along $\mathbf{r}_u$ and $\mathbf{r}_v$ must vanish!

Theorem 8.6 (Geodesic Differential Equations): A curve $u(s), v(s)$ is a geodesic if and only if it satisfies the coupled non-linear ODEs:

By standard ODE existence and uniqueness theory, given any initial point $p \in S$ and initial unit tangent direction $\mathbf{v} \in T_p S$, there exists a unique geodesic starting at $p$ with initial velocity $\mathbf{v}$!

3. Clairaut's Relation on Surfaces of Revolution

For a surface of revolution $\mathbf{r}(u, v) = (r(u)\cos v, r(u)\sin v, z(u))$ where $r(u)$ is the distance to the axis of rotation:

Theorem 8.7 (Clairaut's Relation, 1735): Along any geodesic on a surface of revolution:

where $r$ is the radius (distance to the rotation axis) and $\alpha$ is the angle made by the geodesic with the meridian curve.

1. Geodesic Curvature $\kappa_g$ and the Darboux Frame

For a curve $C: \mathbf{r}(s)$ on a surface $S$, consider the Darboux frame $\{\mathbf{T}, \mathbf{V}, \mathbf{n}\}$:

• $\mathbf{T} = \mathbf{r}'(s)$ is the unit tangent vector.
• $\mathbf{n}$ is the surface unit normal.
• $\mathbf{V} = \mathbf{n} \times \mathbf{T}$ is the intrinsic normal lying in the tangent plane $T_p S$.

The acceleration vector decomposes as:

where:

• $\kappa_n = \mathbf{r}''(s) \cdot \mathbf{n}$ is the normal curvature.
• $\kappa_g = \mathbf{r}''(s) \cdot \mathbf{V} = \det(\mathbf{T}, \mathbf{r}'', \mathbf{n})$ is the geodesic curvature.

By the Pythagorean theorem:

where $\kappa$ is the space curvature of the curve. Geodesic curvature $\kappa_g$ measures how much the curve curves within the surface itself. A curve is a geodesic if and only if $\kappa_g \equiv 0$.

2. Liouville's Formula for Geodesic Curvature

In an orthogonal coordinate system ($F = 0$): Let $\alpha$ be the angle that the unit tangent $\mathbf{T}$ makes with the $u$-coordinate curve ($\mathbf{r}_u / \sqrt{E}$).

Theorem 8.8 (Liouville's Formula):

3. The Local Gauss-Bonnet Theorem

The Gauss-Bonnet theorem connects the differential geometry of curvature directly to the global topology of the surface!

Theorem 8.9 (Local Gauss-Bonnet Theorem): Let $T \subset S$ be a simply connected region (e.g., a curvilinear geodesic triangle) bounded by a piecewise smooth curve $\partial T$ with exterior angles $\epsilon_1, \epsilon_2, \dots, \epsilon_k$ at the corners. Then:

Corollary 8.1 (Geodesic Triangles): If $T$ is a geodesic triangle (its three boundary edges are geodesics, so $\kappa_g = 0$ along all edges) with interior angles $\alpha_1, \alpha_2, \alpha_3$: Since exterior angles are $\epsilon_i = \pi - \alpha_i$:

The integral of Gaussian curvature over the triangle equals the angular excess $\Delta = \sum \alpha_i - \pi$!

• On a sphere ($K > 0$): sum of angles $> \pi$.
• On a plane ($K = 0$): sum of angles $= \pi$.
• On a pseudosphere ($K < 0$): sum of angles $< \pi$.