Differential Geometry

Introduction

Differential geometry is the tool we use to understand how to adapt concepts such as the distance between two points, the angle between two crossing curves, or curvature of a plane curve, to a surface. For example, if you live on a sphere, you cannot go from one point to another by a straight line while remaining on the sphere. So how do you find the shortest distance from one point to another? Another example: how do we characterise the difference between a cylinder, which you can make by bending a plane without stretching it, and a sphere, which you cannot? For a very readable introduction to the history of differential geometry, see D. J. Struik's account.

 

Curvature

Curvature is an important notion in mathematics, studied extensively in differential geometry. Intuitively, curvature describes how much an object deviates from being "flat" (or "straight" if the object is a line). For a nice overview of the history of the study of curvature, see Michael Garman and Jessica Bonnie's paper. For a fun introduction to curvature in the real world, see this excellent Wired article. Curvature is split into two types: "intrinsic" and "extrinsic". To visualise the difference between intrinsic and extrinsic curvature, take a piece of paper. It can be bent, rolled, cut, and folded (but not stretched) to form surfaces such as a cone, Möbius band, or cylinder. This is possible because these surfaces all have zero Gaussian curvature; they are called "developable surfaces". Note that such surfaces have the same intrinsic (Gaussian) curvature, but have very different extrinsic curvature. Intuitively, a cylinder is clearly in some sense more curved than a flat piece of paper: they have different extrinsic curvature. Moreover, we cannot deform a sheet of paper to form intrinsically curved surfaces such as spheres, without stretching the paper. This gives insight into why map projections of the Earth are often quite unsatisfying.

 

Intrinsic Curvature

"Intrinsic curvature" describes the curvature at a point on a surface, and is independent of how the surface is embedded in space (i.e. it is "intrinsic" to the surface), depending only on distances "as measured within the surface". To measure distances within the surface, we define a metric (i.e. a measure of distance) using geodesics (we explore geodesics in more detail below). To better understand this, consider a sphere. The distance between two points on a sphere is now measured as the length of string needed to join them by holding the string taut across the surface, rather than just measuring through the middle of the sphere.

Gaussian curvature is the most commonly studied intrinsic measure of curvature. We shall discuss this in more detail below. This is a reasonably advanced notion in differential geometry, applying also for manifolds (extensions of surfaces with dimension greater than two). Note that for a 1-dimensional manifold (e.g. a line, curve, circle) there is no intrinsic curvature, only extrinsic curvature.

In physical cosmology, intrinsic curvature is key to understanding the shape of the universe. Physicists study intrinsic rather than extrinsic curvature, since we can only measure distances within the universe rather than in a space outside of reality.

In higher dimensions (greater than two), the notion of curvature is too complicated to describe by a single (scalar) number. In this case tensors describe the curvature, as pioneered by Riemann. For more information on intrinsic curvature, have a read through Antonio Naveira's paper [1].

 

Extrinsic Curvature

"Extrinsic curvature" is a more familiar notion, and historically was the first to be studied of the two types of curvature. Extrinsic curvature of a surface depends on how it is embedded within a space. Examples of extrinsic measures of curvature include geodesic curvature, mean curvature, and principal curvatureNow let us explore some notions of curvature for surfaces, beginning with principal curvature:

 

Principal Curvature, Umbilic Points, and Lines of Curvature

Given a point $p$ on a surface $M$, consider rotating the surface such that  the tangent to $p$ is horizontal, with "up" pointing in a direction orthogonal to the surface. Each vertical plane through $p$, denoted by $E$, meets the surface in a curve $\gamma_{E}$ - see the image below. On $\gamma_{E}$, at the point $p$ there is an "osculating circle" which best approximates the curve. One can understand the osculating circle by considering a car travelling along a curve such as $\gamma_{E}$. If as it passes through $p$ the steering wheel locks, then the circle around which it would continue to travel is our osculating circle (if the car is travelling straight ahead, our "circle" is a straight line, which can be thought of as a circle of infinite radius). The radius $r$ of this circle defines curvature $\kappa_{E} :=  \frac{1}{r}$ on $\gamma_{E}$, with a smaller radius corresponding to greater curvature (and a straight line corresponding to zero curvature). This curvature is attributed a positive sign in accordance with the "side" of the surface on which it lies (this is linked to the orientation of the surface).

Principal Curvature - Copyright © 2006 Eric GabaLines of Curvature near an Umbilical Point

(Copyright © 2006 Eric Gaba)

If $\kappa_{E}$ is constant for all planes through a point $p$ then $p$ is an "umbilic point". Such points may be identified on some of our models of quadric surfaces. These umbilic points can be recognised as having what resembles the above configuration of lines of curvature. At a non-umbilic point, there are planes $E_{1}, E_{2}$ with minimal and maximal curvature respectively. It turns out that $E_{1}$ and $E_{2}$ are always orthogonal, and these planes determine the "principal directions" on a surface, with $\kappa_{E_{1}}, \kappa_{E_{2}}$ denoting the "principal curvatures" at $p$. A curve on $M$ is a "line of curvature" if at all points its tangent vector (i.e. its direction) coincides with a principal direction at that point. Many of our models exhibit lines of curvature on various surfaces. See here for some history on the study of lines of curvature [2].

 

Gaussian Curvature

Gaussian curvature is defined as the product $\kappa_{E_{1}} \kappa_{E_{2}}$ of the principal curvatures at a point. Remarkably, it is an intrinsic measure, although it is defined in terms of extrinsic properties. The fact that Gaussian curvature is intrinsic is the result of Gauss's renowned "Theorema Egregium", one of the most important results in differential geometry. Gaussian curvature is perhaps the most natural measure of curvature, and with our models we will mostly be looking at Gaussian curvature rather than other measures of curvature.

Flat surfaces (such as the plane $\mathbb{R}^2$) have zero Gaussian curvature. When Gaussian curvature is positive, the surface can locally be thought of as "like a sphere". Imagine cutting out a small piece of a rubber sphere, and trying to press it flat against a flat surface. This is impossible without cutting or stretching either surface. Negative curvature corresponds to a "saddle-like" picture, which is exhibited on the hyperbolic paraboloid below. As with positive Gaussian curvature, it is impossible to press a negatively curved surface to a surface with a different value of Gaussian curvature without breaking one of them.

Model III 14

 

Geodesics and Geodesic Curvature

Geodesics generalise the notion of a straight line (or line segment) in Euclidean geometry to more general surfaces and spaces. A related notion is that of  "geodesic curvature". For a curve drawn on a surface, its geodesic curvature measures how much it deviates from being a geodesic (i.e. how 'bendy' or 'wiggly' it is within the surface). Geodesics have zero geodesic curvature, and are best understood as curves which locally minimise length. The study of geodesics, has far-reaching applications. For example, constructing frames for aeroplanes using intersecting geodesics yields incredibly strong structures, capable of withstanding much damage while staying airworthy. Such is the power of mathematics. Also geodesics can make free-form architectural structures possible [3]. Like much of differential geometry, geodesics are of interest to physicists, especially in general relativity. "Geodesy" is the study of the measurement and representation of the Earth, with a broad and intriguing history. As the name suggests, geodesics play a key role in geodesy. 

 

Manifolds (more advanced!)

A "manifold" generalises the notion of a surface to higher dimensions. Surfaces are two-dimensional manifolds which live in a three-dimensional space, whereas manifolds can have any dimension, contained in a higher-dimensional space. Familiar examples of 1-dimensional manifolds include circles, lines, curves, and knots

Manifolds are the central object of study in differential geometry, so much so that differential geometry can in fact be characterised as the study of manifolds. Manifolds are essentially "topological spaces" which locally resemble Euclidean space. A topological space is a set of points endowed with a "topology". The topology of the set tells us about its properties which are invariant under deformations such as bending and stretching, but not "gluing" or tearing. The canonical fact used to demonstrate topological properties is the equivalence between a coffee mug and a donut.

More precisely, an "$n$-dimensional manifold" is characterised as locally homeomorphic to $\mathbb{R}^n$ (ordinary $n$-dimensional Euclidean space). This means that if you look at a small enough neighbourhood of any point on the manifold, it will be "smooth", roughly resembling a piece of Euclidean space (but the neighbourhood can be curved).

Simple 1-dimensional examples include curves in the plane such as circles, conic sections, and graphs of smooth functions. Consider a small piece of a curve, as shown below. It clearly resembles $\mathbb{R}$ (the real line) in some sense, but is slightly curved.

Quadratic CurveTangent Plane to a Sphere

Higher dimensional examples include spheres (2-dimensional), smooth surfaces (2-dimensional), and for higher dimensions non-trivial manifolds include 3-tori (3 dimensions) and hyperspheres ($n$ dimensions). Some models in our collection represent manifolds, specifically smooth algebraic surfaces in $\mathbb{R}^3$ (2-dimensional manifolds).

Mathematicians often work with a very powerful and important type of manifold, called a Riemannian manifold. These are manifolds, equipped with a special inner product operator (think of this as a generalisation of the familiar dot product) on their tangent space at each point (think of this as generalising the notion of a tangent plane to a surface, seen above). We will not worry too much about the details here.

References

[1] A. M. Naveira, The Riemann Curvature Through History (2005), http://www.researchgate.net/publication/228958530_The_Riemann_Curvature_...

[2] JORGE SOTOMAYOR, HISTORICAL COMMENTS ON MONGE’S ELLIPSOID AND THE CONFIGURATIONS OF LINES OF CURVATURE ON SURFACES IMMERSED IN R3, arXiv:math/0411403v1

[3] Claudio Pirazzi and Yves Weinand, Geodesic Lines on Free-Form Surfaces - Optimized Grids for Timber Rib Shells, http://infoscience.epfl.ch/record/118623/