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Differential Geometry

branch of mathematics that studies the geometry of curves, surfaces, and manifolds (the higher-dimensional analogues of surfaces). It is called differential geometry because traditionally DG has used the ideas and techniques of calculus, but modern DG often uses algebraic and purely geometric techniques instead of calculus. Although basic definitions, notations, and analytic description vary widely, the following geometric questions prevail: How does one measure the curvature of a curve within a surface (intrinsic) versus within the encompassing space (extrinsic)? How can the curvature of a surface be measured? What is the shortest path within a surface between two points on the surface? How is the shortest path on a surface related to the concept of a straight line? Rigorous answers to these questions, involving techniques from calculus, differential equations, algebra, and other areas are beyond the scope of this article. Instead, we present an informal and intuitive introduction to the main concepts, interspersed with some motivational history.

Introduction

The discovery of calculus opened up the study of complicated plane curves — such as those Descartes produced with his "compass" (see history of geometry) — beyond the simpler curves of antiquity. The methods of calculus, in particular integration, led to the solution of the ancient problems of finding the arc length of a plane curve and of finding the area of a plane figure. This opened the stage to the investigation of curves and surfaces in space — it is this investigation that was the start of differential geometry.

Example: Strakes and "spiral" staircases

The strake and helical staircase railing: To give structural support to large metal cylinders, such as large smoke stacks, engineers sometimes attach a spiraling strip called a strake. (See Figure 1: Strake and helical "spiral" stair railings.) Similarly, the strake shape may be used as the hand rail for a "spiral" (more properly called "helical") staircase or the trim along the top of the stair treads. To produce the strake it is convenient to cut annular strips (the region between two concentric circles) from a flat sheet of steel (plywood or similar material) as illustrated in the figure. These annular pieces are then bent along a helix on the cylinder to form the strake.

Figure 1: Strakes and "Spiral" staircases

But what should the radius r of the annulus be, in order to make the best fit? This is answered in DG by defining the curvature of a curve and then adjusting r until the curvature of the inside edge of the annulus is equal to the curvature of the helix.

And, will the annular strip fit without stretching onto the strake around the cylinder? It will certainly have to be bent, bending (such as in bending a sheet a paper to form a half-cylinder) is OK. In particular, we want that no distances (as measured along the surfaces) will change when we bend the annular strip into the strake. Two surfaces are said to be isometric if they can be bent (or transformed in anyway) one onto the other without changing distances as measured along the surfaces. Thus the second question becomes: Are the annular strip and the strake isometric? To answer this and similar questions DG developed the notion of the curvature of a surface.

The strake and related surfaces are common surfaces in the physical world — for example, if we make the strake very wide compared to the diameter of the cylinder, then we get an auger or spiral parking ramp. If we double the auger (extend it in both directions from the cylinder) and then shrink the cylinder to zero radius, the resulting figure is called a helicoid. (See Figure 2.)

Auger. Helicoid.
Figure 2:

Curvature of curves

Although mathematicians from antiquity had described some curves as curving more than others and straight lines as not curving at all, Gottfried Leibniz (in 1686) was the first to define the curvature of a curve at each point in terms of the circle that best approximates the curve at that point, this circle he called the osculating circle (latin, osculare, to kiss) (see the figure). He then defined the curvature of the curve (and the circle) as 1/r, where r is the radius of the osculating circle. As a curve becomes straighter, a circle with a larger radius must be used to approximate it and the resulting curvature decreases. In the limit, a straight line is said to be equivalent to a circle of infinite radius and its curvature defined as zero everywhere. The only curves with constant curvature are a straight line, a circle, or a helix. In practice, the numerical curvature is found with a formula (discovered by Newton and Leibniz for plane curves and by Euler (1736) for curves in space) which gives the rate-of-change (derivative) of the tangent to the curve as one moves along the curve.

For the study of space curves, the curvature alone is not enough; in addition one needs to know at each point the plane of the osculating circle (which is also the plane most closely containing the curve at the point) and the rate (called the torsion) at which the curve is leaving that plane. These were first studied independently about 1850 by F. Fernet and J.A. Serret. The helix is the only curve with constant non-zero torsion (and constant curvature).

Now we can compute the ideal inner radius r of the annular strip that goes into making the strake or helical hand rail. (See Figure 1.) The curvature of the inner arc of the annular strip, 1/r, we want to be equal to the curvature of the helix on the cylinder. If R is the radius of the cylinder and H is the height of one turn of the helix, then the curvature of the helix can be computed to be 4p2R/[H2+(2pR)2]. In the case of R = 1m and H = 10m, this gives us r = 3.533m.

Curvature of Surfaces

To measure the curvature of a surface at a point p, Euler (in 1760) looked at cross-sections of the surface made by planes which contained the line perpendicular (or "normal") to the surface at p. (See figure: illustrating a cylinder and a sphere cut by various planes containing a normal to the cylinder or sphere) Euler called the curvatures of these cross-sections the normal curvatures of the surface at p. For example, on a vertical cylinder the vertical cross-sections are straight and thus have zero curvature and the horizontal cross-sections are circles which have curvature 1/r. The normal curvatures at a point on a surface are different in different directions. The maximum and minimum normal curvatures at a point p on a surface are called the principle (normal) curvatures and the directions in which these normal curvatures occur are called the principle directions. Euler proved that for most surfaces where the normal curvatures are not constant (for example the cylinder), these principle directions are perpendicular to each other. (Note that on a sphere all the normal curvatures are the same and thus all are principle curvatures.) These principle normal curvatures are a measure how much the surface is curving.

The theory of surfaces and principle normal curvatures was extensively developed by the French Geometry school led by Gaspard Monge (1746-1818). But it was in an 1827 paper that C.F. Gauss made the big breakthrough that allows DG to answer the question of whether the annular strip can be made into the strake without distortion. The (Gaussian) curvature of a surface at the point p is defined to be the product of the two principle normal curvatures and to be positive if two principle normal curvatures curve in the same direction (as on a sphere) and negative if the two principle normal curvatures curve in opposite directions. (See figure [Principle curvatures and Gaussian curvature].) A plane surface has all its normal curvatures zero, and thus the Gaussian curvature of a plane is 0. For a cylinder, the minimum normal curvature is zero (along the vertical straight lines) and the maximum is 1/r, r is the radius of the cylinder and thus the Gaussian curvature is also zero. If we cut the cylinder along one of the vertical straight lines the resulting surface can be flattened (without stretching) onto a rectangle in the plane. (see figure [cylinder and plane are locally isometric].) In DG, we say that the plane and cylinder are locally isometric. These are special cases of two important theorems:

Gauss' Theorema Egregium (Remarkable Theorem) (1827). If two smooth surfaces are isometric then the two surfaces have the same Gaussian curvature at corresponding points. (That is, Gaussian curvature is an intrinsic notion though defined extrinsically.)

Theorem (Minding 1839). Two smooth surfaces with the same constant Gaussian curvature are locally isometric.

Corollaries.

The Gaussian curvature of an annular strip (being in the plane) is constantly zero. So to answer whether or not the annular strip can be isometric to the strake we need only check whether a strake has constant zero Gaussian curvature. In fact, it can be calculated that Gaussian curvature of the strake is negative and not zero. Thus in making a strake or helical hand rail there must be some stretching, but if the strake is narrow the amount of stretching is very small. Sidebar?[This can be seen by looking at an auger which is an extended strake (see figure: normal curvatures on strake and auger). The horizontal straight line segments are normal cross-sections on the strake with zero curvature; but if we look at cross-sections to either side of these they will curve up on one side and curve down on the other and thus the Gaussian curvature must be negative.]

What are shortest paths on a surface?

From our outside, or extrinsic, point-of-view no curve is straight on a sphere -- they all have (extrinsic) curvature. However, the great circles are intrinsically straight: From the point-of-view of an ant crawling along a great circle, there will be no turning or curving with respect to the surface. (see spherical geometry.) Ferdinand Minding (1806-85, taught at Dorpat, now Tartu in Estonia) defined in about 1830 a curve on a surface to be a geodesic if it is intrinsically straight; that is, there is no curvature identifiable from within the surface. It is one of the major tasks of DG to determine what are the geodesics on a surface. The great circles (being circles) do extrinsically have curvature but the curvature is in the direction of the center of the sphere and thus can not be experienced intrinsically. The great circles are the geodesics on a sphere.

A great circle arc that is longer than a half circle is (intrinsically) straight on the sphere but is not the shortest distance between its endpoints. On certain cones (those with cone angles greater than 360°) the shortest paths are not always straight. (See Figure 3: Shortest is not straight!)

Figure 3. Shortest paths that are not straight, and Straight paths are not shortest.

However, an important theorem is:

On a surface which is complete (every geodesic can be extended indefinitely) and smooth then every shortest curve is intrinsically straight and every intrinsically straight curve is the shortest curve between nearby points.