First and Second Fundamental Forms

In differential geometry, we study the properties of curves and surfaces embedded in Euclidean space. To understand the local geometry of a surface, we use fundamental forms. These forms help us measure distances and angles on the surface itself, without reference to the ambient space.

The First Fundamental Form

The first fundamental form, also known as the metric tensor or the first fundamental coefficients, provides a way to measure lengths and angles on a surface. Consider a surface S parameterized by $\mathbf{r}(u, v) = (x(u, v), y(u, v), z(u, v))$. The tangent vector to a curve on the surface is given by $\mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial u}$ and $\mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial v}$.

The first fundamental form is a quadratic form that describes the dot product of tangent vectors. It is expressed as:

$ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2$

where $E$, $F$, and $G$ are the coefficients of the first fundamental form, defined as:

$E = \mathbf{r}_u \cdot \mathbf{r}_u = ||\mathbf{r}_u||^2$

$F = \mathbf{r}_u \cdot \mathbf{r}_v$

$G = \mathbf{r}_v \cdot \mathbf{r}_v = ||\mathbf{r}_v||^2$

These coefficients are functions of $u$ and $v$. The first fundamental form allows us to calculate the length of any curve on the surface and the angle between any two curves intersecting at a point on the surface. For instance, the arc length $s$ of a curve $\gamma(t) = \mathbf{r}(u(t), v(t))$ from $t_1$ to $t_2$ is given by:

$s = \int_{t_1}^{t_2} \sqrt{E \left(\frac{du}{dt}\right)^2 + 2F \frac{du}{dt}\frac{dv}{dt} + G \left(\frac{dv}{dt}\right)^2} \, dt$

The first fundamental form essentially defines the intrinsic geometry of the surface – the geometry that can be measured by an ant living on the surface without seeing the outside world.

The Second Fundamental Form

While the first fundamental form deals with the intrinsic geometry (lengths and angles), the second fundamental form captures how the surface is curved in the ambient Euclidean space. It measures the change in the surface's normal vector as we move along the surface.

Let $\mathbf{n}$ be the unit normal vector to the surface at a point $\mathbf{r}(u, v)$. The second fundamental form is a quadratic form that relates the second-order differentials of the surface coordinates to the dot product of tangent vectors with the normal vector. It is expressed as:

$II = L \, du^2 + 2M \, du \, dv + N \, dv^2$

where $L$, $M$, and $N$ are the coefficients of the second fundamental form, defined using the partial derivatives of the normal vector and the surface parameterization:

$L = \mathbf{r}_{uu} \cdot \mathbf{n}$

$M = \mathbf{r}_{uv} \cdot \mathbf{n}$

$N = \mathbf{r}_{vv} \cdot \mathbf{n}$

Here, $\mathbf{r}_{uu} = \frac{\partial^2 \mathbf{r}}{\partial u^2}$, $\mathbf{r}_{uv} = \frac{\partial^2 \mathbf{r}}{\partial u \partial v}$, and $\mathbf{r}_{vv} = \frac{\partial^2 \mathbf{r}}{\partial v^2}$.

The second fundamental form helps us understand the extrinsic curvature of the surface. For example, it is used to define the normal curvature, which is the curvature of a normal section curve.

The relationship between the two fundamental forms is crucial. The first fundamental form tells us how to measure lengths and angles on the surface, while the second fundamental form tells us how the surface bends in space.

Lines of Curvature

Lines of curvature are special curves on a surface that characterize its bending. At any point on a smooth surface (excluding umbilic points), there are generally two principal directions. These directions correspond to the directions of maximum and minimum normal curvature. Lines of curvature are curves on the surface that lie along these principal directions at every point.

Formally, a curve $\gamma(t) = \mathbf{r}(u(t), v(t))$ on the surface is a line of curvature if its acceleration vector, when projected onto the tangent plane, is parallel to the tangent vector of the curve. In terms of the fundamental forms, this means that the normal vector $\mathbf{n}$ to the surface is parallel to the acceleration vector of the curve in the ambient space.

Mathematically, a curve is a line of curvature if the tangent vector $\mathbf{r}'(t)$ is an eigenvector of the shape operator (also known as the Weingarten map) $S$, where $S(\mathbf{v}) = -\nabla_{\mathbf{v}} \mathbf{n}$. The shape operator relates the change in the normal vector to the tangent vector.

The condition for a curve to be a line of curvature can also be expressed using the coefficients of the fundamental forms. A curve $\gamma(t) = \mathbf{r}(u(t), v(t))$ is a line of curvature if at every point, the tangent vector $\mathbf{r}'(t)$ is proportional to the normal vector of the surface's osculating plane:

$\mathbf{r}''(t) = \kappa \mathbf{N}(t)$

where $\kappa$ is the curvature and $\mathbf{N}(t)$ is the principal normal to the curve. For a line of curvature, the acceleration vector $\mathbf{r}''(t)$ lies in the direction of the surface normal $\mathbf{n}$ or is zero.

A more practical condition involves the coefficients of the fundamental forms. A curve $\gamma(t) = \mathbf{r}(u(t), v(t))$ is a line of curvature if:

$(L \, du + M \, dv) \, du + (M \, du + N \, dv) \, dv = 0$

This equation can be rewritten as:

$L \left(\frac{du}{dt}\right)^2 + 2M \frac{du}{dt}\frac{dv}{dt} + N \left(\frac{dv}{dt}\right)^2 = 0$

This is a quadratic equation in the ratio $du/dv$. The solutions give the directions of the lines of curvature.

The principal curvatures $\kappa_1$ and $\kappa_2$ are the curvatures of the normal sections in the directions of the lines of curvature.

Lines of curvature are important because they provide a natural coordinate system on a surface. They are the curves along which the surface bends most strongly or least strongly.

Meusnier's Theorem

Meusnier's theorem relates the curvature of a curve lying on a surface to the curvature of the normal section in the same tangent direction. It provides a fundamental link between the extrinsic curvature of a curve and the intrinsic geometry of the surface it lies on.

The theorem states that if $\gamma$ is a curve lying on a surface $S$, and $\mathbf{v}$ is the unit tangent vector to $\gamma$ at a point $P$, then the curvature $\kappa$ of $\gamma$ at $P$ is related to the normal curvature $\kappa_n$ of the surface in the direction $\mathbf{v}$ by:

$\kappa \cos \theta = \kappa_n$

where $\theta$ is the angle between the unit normal vector $\mathbf{n}$ to the surface $S$ at $P$ and the unit principal normal vector $\mathbf{N}$ to the curve $\gamma$ at $P$.

Let's break this down:

• Curve $\gamma$: A curve lying on the surface $S$.
• Tangent vector $\mathbf{v}$: A unit vector tangent to $\gamma$ at point $P$.
• Curvature $\kappa$: The magnitude of the curvature vector of $\gamma$ at $P$. This measures how sharply the curve bends.
• Normal vector to the surface $\mathbf{n}$: A unit vector perpendicular to the surface $S$ at $P$.
• Principal normal vector to the curve $\mathbf{N}$: A unit vector in the direction of the curvature of $\gamma$ at $P$.
• Angle $\theta$: The angle between $\mathbf{n}$ and $\mathbf{N}$.
• Normal curvature $\kappa_n$: The curvature of the normal section of $S$ at $P$ in the direction $\mathbf{v}$. A normal section is the curve formed by the intersection of the surface $S$ with a plane containing the normal vector $\mathbf{n}$ and the tangent vector $\mathbf{v}$.

The term $\cos \theta$ represents the projection of the curve's principal normal vector $\mathbf{N}$ onto the surface normal vector $\mathbf{n}$. If $\mathbf{N}$ is aligned with $\mathbf{n}$ (i.e., $\theta = 0$), then $\cos \theta = 1$, and $\kappa = \kappa_n$. This happens when the curve's bending is entirely in the normal direction of the surface, meaning the curve is a normal section.

If the curve's tangent vector $\mathbf{v}$ is in the direction of the surface normal $\mathbf{n}$ (which is impossible for a curve lying *on* the surface, but conceptually), or if the curve is tangent to a line of curvature, the relationship simplifies.

Meusnier's theorem is significant because it shows that the curvature of any curve on a surface at a point $P$ in a given direction $\mathbf{v}$ is determined by the normal curvature of the surface in that same direction. This means we only need to study the normal curvatures to understand the bending properties of the surface.

Gaussian Curvature

Gaussian curvature, denoted by $K$, is a fundamental measure of the intrinsic curvature of a surface at a point. It was introduced by Carl Friedrich Gauss and is invariant under isometries (transformations that preserve distances). This means that if you bend a surface without stretching or tearing it, the Gaussian curvature at each point remains the same.

Gaussian curvature is defined using the coefficients of the first and second fundamental forms:

$K = \frac{LN - M^2}{EG - F^2}$

where $E, F, G$ are the coefficients of the first fundamental form and $L, M, N$ are the coefficients of the second fundamental form.

The sign of the Gaussian curvature tells us about the local shape of the surface:

• $K > 0$ (Elliptic point): The surface is locally like a convex dome. All normal curvatures have the same sign. Examples include spheres and ellipsoids.
• $K < 0$ (Hyperbolic point): The surface is locally like a saddle. Normal curvatures have opposite signs. Examples include hyperboloids and hyperbolic paraboloids.
• $K = 0$ (Parabolic point): The surface is locally like a cylinder or a plane. One of the principal curvatures is zero. Examples include cylinders and planes.
• $K = 0$ and principal curvatures are zero (Planar point): The surface is locally flat.

Gaussian curvature can also be expressed in terms of the principal curvatures, $\kappa_1$ and $\kappa_2$, which are the eigenvalues of the shape operator (or equivalently, the roots of a characteristic equation derived from the fundamental forms).

$K = \kappa_1 \kappa_2$

This means the Gaussian curvature is the product of the maximum and minimum normal curvatures at a point.

Another important result related to Gaussian curvature is the Gauss-Bonnet theorem, which relates the integral of Gaussian curvature over a compact surface to its Euler characteristic.

For a surface given by $\mathbf{r}(u, v)$, the Gaussian curvature can also be computed using the formula:

$K = \frac{\det(\mathbf{r}_{uu}, \mathbf{r}_{uv}, \mathbf{n})}{\det(\mathbf{r}_u, \mathbf{r}_v, \mathbf{n})^2}$ (This is a simplified conceptual representation, the actual calculation involves cross products and dot products).

A more direct formula for $K$ in terms of partial derivatives of $\mathbf{r}$ is:

$K = \frac{\det \begin{pmatrix} \mathbf{r}_{uu} \cdot \mathbf{n} & \mathbf{r}_{uv} \cdot \mathbf{n} \\ \mathbf{r}_{uu} \cdot \mathbf{n} & \mathbf{r}_{vv} \cdot \mathbf{n} \end{pmatrix}}{(\mathbf{r}_u \times \mathbf{r}_v) \cdot (\mathbf{r}_u \times \mathbf{r}_v)}$

Which simplifies to:

$K = \frac{LN - M^2}{EG - F^2}$

The quantity $EG - F^2$ is the determinant of the metric tensor matrix, which is positive for a regular surface.

Euler's Theorem

Euler's theorem in differential geometry relates the normal curvature of a surface in any given direction to the principal curvatures. It provides a way to calculate the normal curvature at a point without needing to find the specific normal section curve.

The theorem states that if $\kappa_1$ and $\kappa_2$ are the principal curvatures at a point $P$ on a surface $S$, and $\theta$ is the angle between the direction of a tangent vector $\mathbf{v}$ and the direction of the tangent vector corresponding to $\kappa_1$, then the normal curvature $\kappa_n$ in the direction $\mathbf{v}$ is given by:

$\kappa_n = \kappa_1 \cos^2 \theta + \kappa_2 \sin^2 \theta$

Let's understand the components:

• Principal Curvatures ($\kappa_1, \kappa_2$): These are the maximum and minimum normal curvatures at a point $P$. They occur in the directions of the lines of curvature.
• Tangent vector $\mathbf{v}$ at $P$: A direction on the tangent plane at $P$.
• Angle $\theta$: The angle between the direction of $\mathbf{v}$ and the direction of the tangent vector associated with $\kappa_1$.
• Normal Curvature ($\kappa_n$): The curvature of the normal section at $P$ in the direction $\mathbf{v}$.

This formula shows that the normal curvature in any direction is a weighted average of the two principal curvatures, with the weights determined by the angle $\theta$.

Special Cases of Euler's Theorem:

• If $\mathbf{v}$ is in the direction of $\kappa_1$ (i.e., $\theta = 0$), then $\cos^2 \theta = 1$ and $\sin^2 \theta = 0$. So, $\kappa_n = \kappa_1(1) + \kappa_2(0) = \kappa_1$.
• If $\mathbf{v}$ is in the direction of $\kappa_2$ (i.e., $\theta = \pi/2$), then $\cos^2 \theta = 0$ and $\sin^2 \theta = 1$. So, $\kappa_n = \kappa_1(0) + \kappa_2(1) = \kappa_2$.
• If $\mathbf{v}$ is exactly halfway between the directions of $\kappa_1$ and $\kappa_2$ (i.e., $\theta = \pi/4$), then $\cos^2 \theta = 1/2$ and $\sin^2 \theta = 1/2$. So, $\kappa_n = \kappa_1(1/2) + \kappa_2(1/2) = (\kappa_1 + \kappa_2)/2$. This is the average of the principal curvatures.

Note that the angle $\theta$ can be measured with respect to either $\kappa_1$ or $\kappa_2$. If measured with respect to $\kappa_2$, the formula would be $\kappa_n = \kappa_1 \sin^2 \theta + \kappa_2 \cos^2 \theta$, where $\theta$ is the angle with $\kappa_2$. However, the standard convention uses the angle with $\kappa_1$.

Euler's theorem is closely related to Meusnier's theorem. Meusnier's theorem states $\kappa \cos \theta = \kappa_n$, where $\theta$ is the angle between the curve's normal and the surface normal. Euler's theorem provides a way to calculate this $\kappa_n$ using principal curvatures.

The relationship between Gaussian curvature ($K$) and mean curvature ($H$) with principal curvatures is:

$K = \kappa_1 \kappa_2$

$H = \frac{\kappa_1 + \kappa_2}{2}$ (The mean curvature is the average of the principal curvatures).

Euler's theorem can be rewritten in terms of $H$ and $K$. If we let $\phi$ be the angle between the direction of $\mathbf{v}$ and the direction of maximum curvature, then $\kappa_n = H + (K/H) \cos(2\phi)$. This form is less common but highlights the relationship with $H$ and $K$. The original form using $\cos^2\theta$ is more standard.

Summary and Interconnections

The first and second fundamental forms are the foundational tools for studying the geometry of surfaces. The first form ($ds^2$) defines distances and angles intrinsically, while the second form ($II$) captures how the surface bends in space.

Lines of curvature are special curves on a surface that align with its principal directions of bending.

Meusnier's theorem connects the curvature of any curve on a surface to the normal curvature of the surface in the same direction.

Gaussian curvature ($K$) is a key intrinsic invariant, telling us about the local shape (elliptic, hyperbolic, parabolic). It's the product of the principal curvatures ($K = \kappa_1 \kappa_2$).

Euler's theorem provides a formula to calculate the normal curvature in any direction using the principal curvatures and the angle of that direction relative to a principal direction ($\kappa_n = \kappa_1 \cos^2 \theta + \kappa_2 \sin^2 \theta$).

These concepts together provide a comprehensive picture of how surfaces curve and bend, both intrinsically and extrinsically.