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ELEMENTARY DIFFERENTIAL GEOMETRY
1-1. Curves
In the first chapters of this book we study plane differential geometry. We start with an investigation of the various definitions of a curve. Our intuitive notion of a "curve" contains so many different features that it is necessary to introduce a number of concepts in order to arrive at an exact definition that is neither too broad nor too narrow for our purposes. We will see that different branches of differential geometry deal with different notions of a curve. A detailed discussion of the various definitions of a curve will also lead to a better understanding of the theory of surfaces and higher-dimensional spaces in later parts of the book.
The idea of a curve which we are trying to formalize is that of a piece of wire that has been twisted and stretched into some odd shape but has not been torn apart. Mathematically, the wire becomes an interval of the real-number line, and the operation performed on it is a continuous map.
Definition 1-1.A plane Peano curve is a continuous map of the unit interval [0,1] into the plane.
We will use the standard notations I for the unit interval and R2 for the plane. A Peano curve may be given in an easily understood shorthand as
f: I -> R2
A point in I, that is, a parameter value 0 ≤ t ≤ 1, has as its image f(t) a point in R2. In some cartesian system of coordinates, the Peano curve is given by an ordered pair of real-valued, continuous functions
f(t) = (x1(t),x2(t))
In at least one important aspect our analytic definition does not agree with the intuitive geometric picture that we want to formalize. A Peano curve defines the set of points covered by it; it is not itself a point set.
Example 1-1. The four curves
f1(t) = (cos 2πt, sin 2πt) f2(t) = (cos 2πt2, sin 2πt2) f3(t) = (sin 2πt, cos 2πt) f4(t) = (cos 4πt, sin 2πt)
are distinct Peano curves. The point set defined in the plane by any one of the four curves is the unit circle S1: x12 + x22 = 1.
From a geometric point of view, the maps f1 and f2 should define the same curve. Under f3, the unit circle is endowed with a negative orientation, and under f4 there are two values t for each point on the circle. This example shows that we need a suitable notion of equivalence of Peano curves, with the understanding that equivalent curves should represent the same "geometric object." The geometric properties of the object will then be those properties of the mapping function f that are common to all equivalent maps. A similar process is used in euclidean geometry where we consider as abstractly identical all congruent figures, although they may differ by their position in the plane. We shall introduce several types of equivalence; we shall identify some curves defining the same point set with different parametrizations; and sometimes we shall identify point sets that may be brought into one another by some motion or transformation of the plane in itself. It will be one of our major results that geometry is the study of invariants of certain "equivalences" or "identifications."
Definition 1-2.A continuous curve defined by a Peano curve f is the set of all maps g(t) = f(H(t)), where H is a one-to-one continuous map of I onto itself, and H(0) = 0.
A one-to-one map H such that both H and its inverse [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are continuous is called a homeomorphism. If H is defined on a compact set (such as I), the continuity of H implies that of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the geometric object "continuous curve f" is uniquely defined by any one of the functions g in the set. Any such admissible function g = fH will be called a representative of the continuous curve. In example 1-1, the maps f1 and f2 represent the same continuous curve, since t -> t2 is a homeomorphism of I onto itself that leaves the origin fixed.
Example 1-2. The continuous curve "segment of y = 2x bounded by the origin (0,0) and the point (1,2), outward orientation," may be represented, for example, by g1(t) = (t,2t) or by g2(t) = (t2,2t2) or by
g3 (t) = (e1-1/t,2e1-1/t) for t > 0, g3 (0) = (0,0)
A continuous curve may also be given by a map f(u), a ≤ u ≤ b, of any closed interval [a,b] into the plane, since u = a + (b - a)t is a one-to-one continuous map of I onto [a,b]. The same remark will hold true for all other classes of curves to be introduced.
A Peano curve is closed if f(0) = f(1). Our definition of a continuous curve implies g(0) = g(1) = f(0) = f(1) for all representatives of a closed curve. This means that we give a special significance to the starting point f(0). For example, the function
f5 (t) = (cos 2π(t + 1/2), sin 2π(t + 1/2)
will not represent the same circle as does f1 (t) in example 1-1, although both describe the unit circle with counterclockwise orientation. To overcome this difficulty, we refer closed curves not to the unit interval I but to the unit circle S1 given by f1 of example 1-1. The parameter t measures the 2π-th part of the arc. We may look at S1 as the image of I in which we have identified the points 0 and 1. A homeomorphism H of S1 onto itself is orientation-preserving if on I it is given by a monotone increasing function.
Definition 1-3.A closed Peano curve is a continuous map of the unit circle S1 into the plane R2. A closed continuous curve defined by a closed Peano curve f is the set of all functions g(t) = f(H(t)), H being an orientation-preserving homeomorphism of S1 onto itself.
According to this definition, both f5 and f1 represent the same circle as a closed continuous curve, the connecting homeomorphism being a rotation of S1 by π. The same map may now represent two distinct geometric beings, viz., a continuous and a closed continuous curve. The two geometric objects may well have different geometric properties.
Actually, we shall use the definitions only for very special continuous curves, since it turns out that any plane continuum (i.e., a compact connected set) can be parametrized so as to become a Peano curve. Peano's famous first example of a pathological "curve" is the parametrization of the entire unit square, obtained in the following way: Divide the square into four parts as shown in Fig. 1-1a. Each smaller square is subdivided into four parts and numbered as shown in Fig. 1-1b. This process is repeated indefinitely. If the real number t ε I is given in its expansion to the base 4, t = a1/4 + a2/42 + ··· + an/4n + ···, an = 0, 1, 2, 3, the image of t is the unique limit point p(t) contained in the nested sequence of squares with labels a1, a1 a2, a1 a2 a3, .... The mapping t ->p(t) is continuous. For any ε > 0, take k such that 4k ? > [square root of 2]. If [absolute value of t* - t] < δ(?) = 1/4k, the first k - 1 digits in the expansions of t and t* coincide; hence p(t*) and p(t) are in the same square of the (k - 1)-th subdivision, and the distance between them is less than the diameter [square root of 2/4k] < ? of such a square.
In one respect, the definitions we have given are too narrow. Many interesting curves are not compact, e.g., a straight line or a logarithmic spiral. Since we are interested in the properties of curves only in the neighborhood of some point, we replace any unbounded curve by a sufficiently large closed segment on it. For example, a line a + bt may be treated through its segments Iλ = {a + bλt}, 0 ≤ t ≤ 1, where λ is a real number.
We will use boldface for vectors and for curves as vector functions. Points in the plane that are not identified with vectors will be denoted by capitals.
A Jordan curve is an equivalence class of homeomorphisms of I into R2 (or of S1 into R2 in the case of closed curves). Though widely used in topology, this notion again is unsuitable for our purposes, as it is both too narrow and too wide — too narrow because it excludes curves with double points; too wide because Jordan curves may not have a tangent at any point. An example of the latter is x(t) = (t,f(t)), where f(t) is a continuous, nondifferentiable function defined on I (see exercise 1-1, Prob. 11). In our final definition, we shall have to include differentiability, and we shall have to find our way between admitting area-filling curves and excluding all singularities.
Definition 1-4.A map f(t) = (x1 (t),x2(t)): I -> R2 is Cn if for any to ε I there exists a neighborhood of t0 in I such that the restriction of f to that neighborhood is a homeomorphism given by n-times continuously differentiable. functions x1 (t), x2 (t). A Cn curve defined by a Cn map f is the set of all functions g(t) = f(H(t)), H being a homeomorphism of I onto _ itself which is an n-times differentiable function, H(0) = 0.
Continuous curves will be called C]degrees] curves. Closed Cn curves are defined similarly from maps S1 -> R2. By the Heine-Borel theorem there is a finite number of neighborhoods covering I so that f is a homeomorphism in each of them. This means that a Cn curve is a finite union of differentiable Jordan curves. Nevertheless, a Cn curve may have an infinity of double points. An example is given at the end of this section. A Cn curve may also have cusps, as is shown by the cycloid
f(t) = (4πt - sin 4πt, 1 - cos 4πt)
The first and second derivatives of the mapping functions have simple geometric interpretations in euclidean geometry. As a consequence, we shall deal mostly with C2 curves. For convex arcs, however, some considerations of curvature involving second derivatives may be circumvented.
Definition 1-5.A set is convex if with any two points it contains the segment defined by the two points (Fig. 1-2a). A curve with endpoints P0 P1, is convex if its point set, together with the segment P0 P1, bounds a convex set in R2 (Fig. 1-2b). A Kn curve is a Cn curve such that for any of its maps f and any t ε I there exists an ? > 0 for which f(t) restricted to [t - ?, t + ?] defines a convex curve.
A Kn curve is the union of a finite number of convex curves, joined together without introducing inflections or cusps. (Inflections and cusps are points at which unions of convex curves cease to be Kn. In a cusp, the "tangent vector" vanishes. The treatment of singularities is avoided in this text.) Such a curve may contain an infinity of double points.
Example 1-3. On a unit circle, denote by Ak the endpoint of the arc of length (1 - 2-k)π measured from (1,0). These points are joined by parabolical arcs having as tangents at Ak the tangents to the circle. The curve obtained in starting from A0, going to A∞ = (-1,0) through the parabolic arcs, returning to A0 by the lower half circle, and then finally to A∞ by the upper half circle is a K1 with an infinity of double points.
Exercise 1-1
1. Is f(t) = (1,0) a Peano curve?
2. A Peano curve f: I -> R2 is given in polar coordinates r(t), θ(t). Show that r(t) and θ(t) are continuous functions if the origin is not a point on the curve.
3. Prove that a real function H(t) defined on I is a homeomorphism of I onto itself, if it is strictly monotone and continuous and if either H(0) = 0, H(1) = 1 or H(0) = 1, H(1) = 0.
4. How many distinct closed continuous curves are given by the mappings of example 1–1?
5. Represent the graph of a continuous function x2 = f(x1), a ≤ x2 ≤ b, as a Peano curve.
6. Show that any homeomorphism of the circle S1 onto itself may be obtained as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where f1 is given in example 1-1, H is a homeomorphism of I onto itself, and R is a rotation of the plane about the origin.
7. Any number m/4n(m,n integers) has two expansions to the base 4, a finite one a1/4 + ··· + an/4n and an infinite one a1/4 + ··· + an-1/4n-1 + (an - 1)/4n + 3/4n+1 + 3/4n+2 + ···. Show that both expansions define the same point on Peano's curve. Draw the approximating squares for p(1/4).
8. Prove that on Peano's curve p(1/6) = p(1/2).
9. Peano's map of I onto the unit square is not one-to-one. Show that any point on a side of a square in Peano's construction will be a double point of the curve, and any point which is a vertex of a square will be a quadruple point.
10. A logarithmic spiral is the graph of r = ce-mθ in polar coordinates. The graph is not a compact set. Given any two points (r0,θ0;r1,θ1) of the spiral, represent by a Peano curve the segment defined on the spiral by the two points.
11. We define a function y(t)(t ε I) by a limit process.
(a) y0 (t) = t.
(b) Assume yn(t) to be defined. We define yn+1 first on the point subdividing I into 3n+1 equal parts. Let
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then define
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In the intervals between these division points the function is defined as the linear function joining the relative values in the division points.
(i) Draw graphs of y0 (t), y1 (t), y2 (t).
(ii) Show that the sequence of functions yn(t) converges to a continuous nowhere-differentiable function y(t).
12. Give a description of the curve of example 1-3 in terms of mapping functions x1(t), x2(t).
13. Give the complete formal definition of a closed Cn (Kn) curve.
14. Find an example of a Kn curve with an infinity of self-intersections.
15. Given two curves f(t), g(t), f(1) = g(0). The curve h(t),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is the sum of f and g. Prove that the point set of h is the union of the point sets of f and g and that h is C0 if both f and g are C0.
16. Find a C0 map I -> R2 which is not C1 and whose image set
{P|f(t) = P}
is a unit circle.
17. A differentiable homeomorphism on an open interval has a differentiable inverse on a closed subinterval. Prove this statement and discuss its application to Def. 1-4.
1-2. Vector and Matrix Functions
It is assumed that the reader is familiar with the fundamentals of vector and matrix algebra. For typographical convenience, a column vector will usually be written as a row in braces:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Row vectors are indicated by parentheses:
{x1,x2, ..., xn}t = (x1,x2, ..., xn)
The symbol t represents the transpose of a matrix.
A system of basis vectors e1, ..., en of an n-dimensional vector space is called a frame in that space and is noted as a column vector of row vectors {e1, ..., en}. Coordinate vectors are row vectors. A vector, i.e., an element of a vector space, is the matrix product of a coordinate vector and a frame:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If the elements of a matrix A = (aij(u)) are differentiable functions of a variable u, the derivative of A is the matrix of the derivatives:
dA/du = (daij/du)
Derivatives with respect to a "general" parameter u will also be written with the "dot" symbol, dA/du = A. The "prime" symbol will be reserved for differentiation with respect to certain "invariant" parameters which we shall introduce later. Such an invariant parameter will often be denoted by s: dA/ds = A'.
Excerpted from Differential Geometry by Heinrich W. Guggenheimer. Copyright © 1977 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Overview
Designed for advanced undergraduate or beginning graduate study, this text contains an elementary introduction to continuous groups and differential invariants; an extensive treatment of groups of motions in euclidean, affine, and riemannian geometry; and development of the method of integral formulas for global differential geometry.