Dupin's Indicatrix

In differential geometry, the Dupin indicatrix is a fundamental concept used to classify the local shape of a surface at a point. It helps us understand whether a point on a surface is locally convex, concave, or saddle-shaped. Imagine a small sphere centered at the point on the surface. We then project the intersection of this sphere with the surface onto the tangent plane at that point. The resulting curve of intersection, when scaled appropriately, forms the Dupin indicatrix.

To define it formally, let $P$ be a point on a surface $S$. Let $T_P(S)$ be the tangent plane to $S$ at $P$. Consider a sphere $\Sigma$ centered at $P$ with a small radius $r$. The intersection of $S$ and $\Sigma$ near $P$ forms a small curve. We can project this curve onto the tangent plane $T_P(S)$. If we consider a specific type of projection related to the curvature of the surface, we obtain the Dupin indicatrix.

More precisely, consider the normal vector $N$ at point $P$. For a point $Q$ on the surface near $P$, let its normal vector be $N_Q$. We are interested in the difference between the normal vector at $P$ and the normal vector at $Q$, projected onto the tangent plane. Alternatively, and more commonly, we consider the intersection of the surface with planes parallel to the tangent plane.

Classification using Dupin's Indicatrix

The shape of the Dupin indicatrix at a point $P$ on a surface $S$ is determined by the principal curvatures, $k_1$ and $k_2$, at $P$. The Dupin indicatrix is an ellipse, a hyperbola, or a parabola.

• Elliptical Point: If both principal curvatures $k_1$ and $k_2$ have the same sign (both positive or both negative), the Dupin indicatrix is an ellipse. This means the surface is locally convex or concave at $P$. For example, the surface of a sphere has elliptical points everywhere.
• Hyperbolic Point: If the principal curvatures $k_1$ and $k_2$ have opposite signs (one positive and one negative), the Dupin indicatrix is a hyperbola. This indicates a saddle shape at $P$. The surface curves up in one direction and down in another. The surface of a saddle is an example of a surface with hyperbolic points.
• Parabolic Point: If at least one of the principal curvatures is zero, the Dupin indicatrix is a parabola. This occurs when the surface is flat in one direction but curved in another. A parabolic cylinder has parabolic points along its line of striction.
• Flat Point: If both principal curvatures are zero ($k_1 = k_2 = 0$), the point is a flat point. This is essentially a planar point, and the Dupin indicatrix degenerates to a point.

The specific shape of the ellipse or hyperbola provides more detailed information about the curvature. The axes of the indicatrix align with the directions of the principal curvatures.

Surfaces of Revolution

A surface of revolution is generated by rotating a curve in a plane around an axis that lies in the same plane. The curve being rotated is called the generating curve or the meridian curve, and the axis of rotation is called the axis of revolution.

Consider a curve $\gamma(t) = (x(t), y(t), 0)$ in the $xy$-plane, and let the axis of revolution be the $z$-axis. When this curve is rotated around the $z$-axis, each point $(x(t), y(t), 0)$ traces a circle in a plane parallel to the $xy$-plane. The surface of revolution $S$ can be parameterized as:

$r(u, v) = (x(u) \cos(v), x(u) \sin(v), y(u))$

Here, $u$ is the parameter along the generating curve, and $v$ represents the angle of rotation around the $z$-axis. For simplicity, we often choose the generating curve to be in the $xz$-plane, so $y(t) = 0$. Let the curve be parameterized by $(x(u), 0, z(u))$. Then the surface of revolution around the $z$-axis is:

$r(u, v) = (x(u) \cos(v), x(u) \sin(v), z(u))$

If we rotate around the $x$-axis, with the generating curve in the $xy$-plane parameterized by $(0, y(u), z(u))$, the surface is:

$r(u, v) = (x(u), y(u) \cos(v), y(u) \sin(v))$

Examples of Surfaces of Revolution

• Sphere: Generated by rotating a semicircle around its diameter. If the semicircle is $x = R \cos(\theta)$, $z = R \sin(\theta)$ for $\theta \in [0, \pi]$ and rotated around the $z$-axis, we get $x^2 + y^2 + z^2 = R^2$.
• Cylinder: Generated by rotating a straight line parallel to the axis of revolution. For example, rotating the line $x = R, z = u$ around the $z$-axis yields $x^2 + y^2 = R^2$.
• Cone: Generated by rotating a straight line that passes through the origin and intersects the axis of revolution. Rotating the line $x = au, z = u$ around the $z$-axis gives $x^2 + y^2 = a^2 z^2$.
• Torus: Generated by rotating a circle around an axis that lies in the same plane as the circle but does not intersect it.

Properties of Surfaces of Revolution

Meridians: The curves on the surface formed by the intersection of the surface with planes containing the axis of revolution are called meridians. These are congruent to the generating curve.

Parallels: The curves formed by the intersection of the surface with planes perpendicular to the axis of revolution are called parallels. These are circles.

The Gaussian curvature and the mean curvature of a surface of revolution have specific formulas related to the curvature of the generating curve. If the generating curve is parameterized by arc length $s$, and its curvature is $\kappa(s)$, and the distance from the axis of revolution is $r(s)$, then the principal curvatures are $k_1 = \kappa(s)$ (the curvature of the meridian) and $k_2 = \frac{\sin \phi}{r(s)}$, where $\phi$ is the angle between the tangent to the generating curve and the parallel circle. A simpler form for $k_2$ is $k_2 = \frac{1}{r(s) \cos \alpha}$, where $\alpha$ is the angle between the tangent to the generating curve and the tangent to the parallel circle.

For a surface of revolution parameterized by $r(u, v) = (x(u) \cos v, x(u) \sin v, z(u))$, the principal curvatures are $k_1$ (curvature of the meridian curve $(x(u), 0, z(u))$) and $k_2 = \frac{x'(u)}{x(u)}$.

Conjugate Systems

A conjugate system on a surface $S$ is a pair of families of curves on $S$ such that each curve in one family intersects each curve in the other family. More importantly, in differential geometry, a conjugate system refers to a pair of cross-curving families of curves, $(u=\text{const}, v=\text{const})$, where the parameter lines are related in a specific way.

Consider a parameterization $r(u, v)$ of a surface. The curves $u = c_1$ and $v = c_2$ form a coordinate net on the surface. A conjugate system is a pair of families of curves, say $\{C_i\}$ and $\{D_j\}$, such that the tangent to a curve $C_i$ at any point $P$ is conjugate to the tangent to a curve $D_j$ passing through $P$, with respect to the quadratic form of the tangent plane.

Let the first fundamental form of the surface be $ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2$. The tangent vector to the curve $u = c$ is $r_v = \frac{\partial r}{\partial v}$, and the tangent vector to the curve $v = c$ is $r_u = \frac{\partial r}{\partial u}$.

Two directions (or tangent vectors) $t_1$ and $t_2$ at a point $P$ on the surface are said to be conjugate with respect to the first fundamental form if $(dr)(t_1, t_2) = 0$, where $dr$ is the first differential quadratic form. If $t_1 = a \, du + b \, dv$ and $t_2 = c \, du + d \, dv$, then $Eac + F(ad+bc) + Gbd = 0$.

For the coordinate net $u=const$ and $v=const$, the tangent directions are $r_v$ and $r_u$. These directions are conjugate if $F = 0$. So, a coordinate net where $F=0$ is called an orthogonal net.

A conjugate system is a pair of families of curves $\{C_\alpha\}$ and $\{D_\beta\}$ such that for any point $P$ on the surface, the tangent to $C_\alpha$ through $P$ and the tangent to $D_\beta$ through $P$ are conjugate directions with respect to the first fundamental form.

A key property is that if a conjugate system exists on a surface, then the asymptotic lines and the lines of curvature form conjugate systems.

Asymptotic Lines and Conjugate Systems

The asymptotic lines on a surface are curves along which the normal curvature is zero. They are the directions for which the second fundamental form $II = L \, du^2 + 2M \, du \, dv + N \, dv^2$ vanishes.

The differential equation for asymptotic lines is $L \, du^2 + 2M \, du \, dv + N \, dv^2 = 0$.

The family of asymptotic lines on a surface forms a conjugate system. This means that if you take a curve from the family of asymptotic lines $C_\alpha$, its tangent at any point $P$ is conjugate to the tangent to another curve $D_\beta$ from a second family of curves passing through $P$. This second family is also formed by asymptotic lines, but often considered in a different "direction" or parameterization.

Specifically, if $u=const$ and $v=const$ define the asymptotic lines, then the tangent to $u=const$ is conjugate to the tangent to $v=const$.

Asymptotic Lines

Asymptotic lines are curves on a surface where the tangent plane to the surface makes zero angle with the normal to the surface in the direction of the curve. More formally, they are curves along which the normal curvature is zero.

Recall that the normal curvature $k_n$ in a direction defined by a unit tangent vector $t$ is given by $k_n = K(t, t)$, where $K$ is the second fundamental form. For a parameterization $r(u, v)$ with the first fundamental form $ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2$ and the second fundamental form $II = L \, du^2 + 2M \, du \, dv + N \, dv^2$, the normal curvature in the direction $(du, dv)$ is:

$k_n = \frac{L \, du^2 + 2M \, du \, dv + N \, dv^2}{E \, du^2 + 2F \, du \, dv + G \, dv^2}$

Asymptotic lines are defined by $k_n = 0$, which means the numerator must be zero:

$L \, du^2 + 2M \, du \, dv + N \, dv^2 = 0$

This is a differential equation that defines the asymptotic lines. At a point where $L, M, N$ are not all zero, this equation can be solved for $du/dv$ or $dv/du$, yielding at most two families of curves passing through the point.

Classification and Properties

• Hyperbolic Points: At a hyperbolic point, the principal curvatures have opposite signs. The second fundamental form $II$ can be factored into two distinct real linear factors, leading to two real distinct families of asymptotic lines passing through the point.
• Elliptic Points: At an elliptic point, the principal curvatures have the same sign. The second fundamental form $II$ is definite (either positive or negative definite). There are no real solutions to $L \, du^2 + 2M \, du \, dv + N \, dv^2 = 0$ unless $L=M=N=0$. Thus, there are no real asymptotic lines passing through an elliptic point.
• Parabolic Points: At a parabolic point, one principal curvature is zero. The second fundamental form $II$ can be factored into two equal real linear factors. This results in one family of asymptotic lines passing through the point.
• Planar Points: At a planar point, both principal curvatures are zero. This means $L=M=N=0$, and the surface is locally flat. Every direction can be considered an asymptotic direction, so the entire surface is covered by asymptotic lines.

Asymptotic lines are important because they represent the "straightest possible" paths on a curved surface in a specific sense, related to how the surface bends. They are also invariant under isometric deformations.

Isometric Lines (Lines of Curvature)

Isometric lines, more commonly known as lines of curvature, are curves on a surface such that their tangent direction at each point coincides with one of the principal directions at that point. The principal directions are the directions in the tangent plane for which the normal curvature is either maximized or minimized.

At any non-umbilic point on a surface, there are two distinct principal directions, corresponding to the maximum and minimum normal curvatures ($k_1$ and $k_2$). The integral curves of these principal directions form two families of curves called the lines of curvature.

The differential equation for the lines of curvature is given by:

$(k_1 - k_n) \, du = 0$ and $(k_2 - k_n) \, dv = 0$ when $F=0$.

A more general form, relating the first and second fundamental forms, is:

$(L - k \, E) \, du + (M - k \, F) \, dv = 0$ $(M - k \, F) \, du + (N - k \, G) \, dv = 0$

where $k$ is a principal curvature. The directions $(du, dv)$ that satisfy these equations for some $k$ are the principal directions. The integral curves of these directions are the lines of curvature.

Properties

• Orthogonality: The two families of lines of curvature are orthogonal to each other, except at umbilic points where $k_1 = k_2$.
• Principal Curvatures: Along a line of curvature, the normal curvature is constant and equal to one of the principal curvatures.
• Surfaces of Revolution: For a surface of revolution, the meridians (curves passing through the axis of revolution) and the parallels (circles perpendicular to the axis) are the lines of curvature.
• Isometric Invariance: Lines of curvature are not generally preserved under arbitrary isometries (transformations that preserve distances). However, they are invariant under certain types of isometries. The term "isometric lines" might be used in some contexts to refer to curves that *are* preserved under isometries, which often relates to asymptotic lines or specific families of curves. However, in the standard curriculum, "lines of curvature" is the term for curves along principal directions.
• Singularities: Lines of curvature can terminate or branch only at umbilic points.

The concept of lines of curvature is crucial for understanding the intrinsic and extrinsic geometry of surfaces. They provide a natural coordinate system on the surface, simplifying many calculations.

Geodesics

Geodesics are the generalization of straight lines to curved surfaces. On a flat plane, a geodesic is simply a straight line segment, representing the shortest path between two points. On a curved surface, a geodesic is a curve that locally minimizes the distance between any two points on the curve. Alternatively, a geodesic is a curve whose tangent vector remains parallel to itself when transported along the curve using parallel transport defined by the surface's geometry.

Mathematically, a curve $\gamma(t)$ on a surface $S$ is a geodesic if its acceleration vector is normal to the surface at every point. If $\gamma(t)$ is parameterized by arc length, its acceleration vector $\gamma''(t)$ is orthogonal to the tangent plane $T_{\gamma(t)}(S)$ for all $t$.

Let the surface be parameterized by $r(u, v)$, and let the curve be given by $u(t), v(t)$. The geodesic equations are derived from the condition that the geodesic curvature is zero. The geodesic curvature $k_g$ is the component of the acceleration vector tangential to the surface. For a curve parameterized by arc length $s$, the geodesic equations are:

$\frac{d^2 u}{ds^2} + \sum_{i,j} \Gamma_{ij}^u \frac{du^i}{ds} \frac{du^j}{ds} = 0$ $\frac{d^2 v}{ds^2} + \sum_{i,j} \Gamma_{ij}^v \frac{du^i}{ds} \frac{du^j}{ds} = 0$

where $\Gamma_{ij}^k$ are the Christoffel symbols of the first kind associated with the first fundamental form $ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2$. The Christoffel symbols are:

$\Gamma_{11}^1 = \frac{1}{2G}(2EG_u - 2FG_v)$ (using $u^1=u, u^2=v$) $\Gamma_{12}^1 = \frac{1}{2G}(2EG_v - 2FG_u)$ $\Gamma_{22}^1 = \frac{1}{2G}(2GG_v - 2FG_u)$ $\Gamma_{11}^2 = \frac{1}{2E}(2EG_u - 2FG_v)$ $\Gamma_{12}^2 = \frac{1}{2E}(2FG_v - 2EG_u)$ $\Gamma_{22}^2 = \frac{1}{2E}(2GG_u - 2FG_v)$

The geodesic equations become:

$\frac{d^2 u}{ds^2} + \Gamma_{11}^1 (\frac{du}{ds})^2 + 2\Gamma_{12}^1 \frac{du}{ds}\frac{dv}{ds} + \Gamma_{22}^1 (\frac{dv}{ds})^2 = 0$ $\frac{d^2 v}{ds^2} + \Gamma_{11}^2 (\frac{du}{ds})^2 + 2\Gamma_{12}^2 \frac{du}{ds}\frac{dv}{ds} + \Gamma_{22}^2 (\frac{dv}{ds})^2 = 0$

These are a system of second-order ordinary differential equations. Given initial position and initial velocity (tangent direction), there exists a unique geodesic passing through that point with that initial direction.

Examples and Properties

• Sphere: Geodesics on a sphere are great circles (circles whose center coincides with the center of the sphere). The shortest path between two points on a sphere is along the shorter arc of the great circle passing through them.
• Cylinder: Geodesics on a cylinder are helices, circles, and straight lines parallel to the axis. If you unroll a cylinder into a plane, these curves become straight lines.
• Cone: Geodesics on a cone, when unrolled onto a plane, become straight lines.
• Shortest Path: Geodesics are locally the shortest paths between points. Globally, they may not be the shortest path (e.g., traveling halfway around a great circle on a sphere).
• Parallel Transport: A curve is a geodesic if and only if its tangent vector is parallel transported along itself. This means the tangent vector doesn't change "intrinsically" as you move along the curve.
• Relationship with Lines of Curvature: On a surface of revolution, the meridians are geodesics. Parallels are generally not geodesics unless they are great circles (which only happens for a cylinder).

The study of geodesics is fundamental to Riemannian geometry and has applications in fields like general relativity, where spacetime is modeled as a curved manifold and the paths of objects under gravity are geodesics.