Lectures on Classical Differential Geometry

Second Edition

Dirk J. Struik

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486138183
• Topic: Mathematics
• Subtopic: Differential Geometry
• Index: Mathematics

Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student.
Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Dr. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there.
For this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. The result was to further increase the merit of this stimulating, thought-provoking text — ideal for classroom use, but also perfectly suited for self-study. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.

CHAPTER 1

CURVES

1–1 Analytic representation. We can think of curves in space as paths of a point in motion. The rectangular coordinates (x, y, z) of the point can then be expressed as functions of a parameter u inside a certain closed interval:

(1–1a)

It is often convenient to think of u as the time, but this is not necessary, since we can pass from one parameter to another by a substitution u = f(v) without changing the curve itself. We select the coordinate axes in such a way that the sense OX → OY → OZ is that of a right-handed screw. We also denote (x, y, z) by (x₁, x₂, x₃), or for short, x_i, i = 1, 2, 3. The equation of the curve then takes the form

(1–1b)

We use the notation P(x_i) to indicate a point with coordinates x_i.

EXAMPLES. (1) Straight line. A straight line in space can be given by the equation

(1–2)

where a_i, b_i are constants and at least one of the b_i ≠ 0.

This equation represents a line passing through the point (a_i) with its direction cosines proportional to b_i. Eq. (1–2) can also be written:

(2) Circle. The circle is a plane curve. Its plane can be taken as z = 0 and its equation can then be written in the form:

(1–3)

FIG. 1–1

Here a is the radius, u = ∠POX (Fig. 1–1).

(3) Circular helix. The equation is

(1–4)

This curve lies on the cylinder x² + y² = a² and winds around it in such a way that when u increases by 2π the x and y return to their original value, while z increases by 2πb, the pitch of the helix (French: pas; German: Ganghöhe). When b is positive the helix is right-handed (Fig. 1–2a); when b is negative it is left-handed (Fig. 1-2b). This sense of the helix is independent of the choice of coordinates or parameters; it is an intrinsic property of the helix. A left-handed helix can never be superimposed on a right-handed one, as everyone knows who has handled screws or ropes.

The functions x_i(u) are not all constants. If two of them are constants Eqs. (1–1) represent a straight line parallel to a coordinate axis. We also suppose that in the given interval of u the functions x_i(u) are single-valued and continuous, with a sufficient number of continuous derivatives (first derivatives in all cases, seldom more than three). It is sufficient for...