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These course-notes are a minor revision (from 2003) of a first
draft that was prepared for a course in spring 2000 at ASU.
Some type-ohs continue to be corrected at irregular times.
They are extremely close to (but nowhere as accurate) as Spi-
vak’s books – and they can be justified only by Spivak be-
ing out of print when the 2000 spring semester began. Since
then Spivak’s volumes have been republished (now typeset,
no longer type-written), and every user of these notes is ex-
pected to eventually buy the originals by Spivak – he deserves
his royalties!
These notes are based on class-notes taken from a class taught
by Al Lundell at the University of Colorado in 1983/84, and
on two classes taught in 1989 and 1991 at Arizona State Uni-
versity. Originally, these notes may have been quite indepen-
dent, but upon efforts to make them more comprehensive,
more precise, they have again converged very much to Spi-
vak’s treatment.
However, in some places the presentation departs from Spi-
vak and material follows more closely e.g. Sternberg (e.g.
notation and terminology involving tensors), Boothby (e.g.
Riemannian basics), Marsden and others. The main practical
value of these notes is that they use the same notation (even
if it is just ui^ in place of xi) that the instructor has become
too accustomed to, and will use in class... and they integrate
questions / exercises for our class.
The current version will include more diagrams – which are es-
sential for readability. Moreover, sections such as the reviews
of basic topology and differentiability, which really belong into
an appendix, are included in the order that the class actually
covered them.
The affiliated explorations that use computer algebra system
have not yet been integrated into these notes.
1 Curves in the plane and 3-space
This first section addresses mostly prerequisite material and is not completely self-contained. It provides some basic definitions and discusses some funda- mental theorems. Central objectives are to raise some questions that will have to be addressed when working in more general settings, and to set the stage for the questions about geometric properties.
1.1 Basic definition of a curve
In many settings it may be appropriate to think of a curve as a set of points in the plane or in 3-space. However, in differential geometry and other advanced settings, it is generally more convenient to work with a different notion – basically calling what previously was named a parameterization the “curve”.
Definition 1.1 A curve is a continuous function defined on an interval I ⊆ R, taking values in a (topological) space M (in this section M is assumed to be Rn). (The interval in this definition may be open or closed, finite, semi-infinite or the entire real line. At this time we only assume enough structure on the space M so that we can talk about continuity.)
The key difference is that with this definition a curve is a function. Consequently it has a richer structure than just a set of points – a structure that facilitates technical analysis. Moreover, this definition easily carries over to much more general settings – e.g. we may think of a vibrating membrane as a curve in an appropriate space M of functions of two variables. What matters is that the space has enough structure (at least a topology) so that we may talk about continuity. (Later we will require additional structures on the space M so that we can differentiate curves.)
Several properties of curves deserve their own names. A curve γ : I 7 → M is called closed if I = [a, b] is a (finite) closed interval and γ(a) = γ(b). If the restriction of a closed curve γ to [a, b) is one-to-one, then γ is called a simple closed curve.
Example 1.1 The circle S^1 = {(x, y) ∈ R^2 : x^2 + y^2 = 1} is the image of the simple closed curve γ : [0, 2 π] 7 → R^2 defined by γ(t) = (cos t, sin t). (Note that the circle S^1 is not a curve!)
Definition 1.2 If γ : I 7 → M is a curve defined on a finite interval I and φ : J 7 → I is a continuous function that maps a finite interval J onto I (mapping endpoints to endpoints), then the curve σ = γ ◦ φ : J 7 → M is called a reparameterization of γ. (Commonly one requires in addition that φ is also one-to-one.)
The notion of reparameterization may be extended to infinite intervals provided one suitably modifies the notion of endpoint to endpoint, e.g. requiring the existence of the limits limt →±∞ and that these equal −∞ and ∞. Among one-to-one reparameterizations one distinguishes orientation-preserving and orientation-reversing reparameterizations according to whether the map φ maps the left endpoint of J to the left or right endpoint of I.
Exercise 1.2 Verify that the curve γ : R 7 → R^2 defined by γ(t) = (cos(t), sin(t)) has constant speed, but nonzero acceleration.
Exercise 1.3 Prove that the acceleration γ′′^ is orthogonal to the velocity γ′^ (for all t), i.e. < γ′(t), γ′′(t) > ≡ 0 if and only if the speed is constant. (Hint: Differentiate c ≡ ‖γ′(t)‖^2 = < γ′(t), γ′(t) >. Read the identities both directions.)
Exercise 1.4 Verify that the curve γ : R 7 → R^2 defined by γ(t) = (t^2 , 0) if t ≥ 0 , and γ(t) = (0, t^2 ) if t < 0 is continuously differentiable, yet its image in the plane has a corner.
Blanket assumption: For most of the following we shall assume that all curves under consideration are twice continuously differentiable and that ‖γ′(t)‖ 6 = 0 for all t.
In many cases we will for convenience even assume that the curve is smooth, i.e. that it has continuous derivatives of all orders. Note that the assumption that the speed is never zero eliminates such nuisances as the corners exhibited by the continuously differentiable curve of exercise 1.4. Moreover, it also prohibits such nuisances as exhibited by the curve γ(t) = (cos(t − t^3 ), sin(t − t^3 )) which “go back and fourth” along the image of the curve. If there is a need to allow for such behaviours it is usually easy to either consider the curve in pieces, or relax the requirements in specific cases and then adapt the desired theorems as needed.
Obviously many different curves may have the same image – and there seems an arbitrariness about picking a specific parameterization. For theoretical purposes it is often convenient to work with a canonical reparameterization of a differentiable curve γ. A natural choice is such that the parameter t (often considered as time) agrees with the distance traveled along the curve.
Definition 1.4 The arc-length of a continuously differentiable curve γ : [a, b] 7 → Rn^ is defined as the function s : [a, b] 7 → R,
s = ψ(t) =
∫ (^) t
a
< γ′(τ ), γ′(τ ) > 1 /^2 dτ (3)
The curve γ is called parameterized by arc-length if ψ(t) ≡ t, or, equivalently, < γ′, γ′^ > ≡ 1.
This definition relies on the standard inner product < · , · > which in Rnis almost synonymous with the notion of (Euclidean) distance. The key idea underlying Riemannian geometry – see chapter 4 – is that once one has a suitable generalization of this inner product, then most of the notions and properties naturally carry over to abstract settings. To explicitly reparameterize a given curve γ : [a, b] 7 → Rn^ by arc-length generally requires one not only to evaluate the integral in equation (3) in closed form, but in addition, to solve the equation s = ψ(t) for t in terms of s – generally a hopeless task if looking for closed form expressions in terms of the traditional elementary functions. Thus one usually has to choose between either formal expressions (for theoretical purposes), and numerical techniques (in practical calculations). We will see that curves that are parameterized by arc-length allow for particular elegant descriptions of their geometry.
Exercise 1.5 Reparameterize the curve γ : [0, 1) 7 → R^2 defined by γ(t) = (
1 − t^2 , t) by arc- length by explicitly integrating equation (3) and solving for t in terms of s.
Exercise 1.6 Calculate and sketch the graphs of the arc-lengths of the curves γ 1 (t) = (t, t), γ 2 (t) = (cos 2πt, sin 2πt), γ 3 (t) = (t, t^2 ), and γ 4 (t) = (cos 2πt, sin 2πt, c t), all defined for t ≥ 0. If feasible, reparameterize each curve by arc-length.
Exercise 1.7 Verify by direct calculation that arc-length of plane curves is invariant under or- thogonal linear transformations: More specifically, let γ : [a, b] 7 → R^2 and σ : [a, b] 7 → R^2 be two differentiable curves that are related by σ = A · γ where A is a 2 × 2 rotation matrix with a 11 = ±a 22 = cos θ and ∓a 12 = a 21 = sin θ for some value of θ ∈ R. Calculate and compare the arc-length functions associated to σ and γ. Repeat for reflections with a 11 = −a 22 = cos θ and a 12 = a 21 = sin θ. Explain in geometric terms – e.g. using the earlier definition in terms of polygonal approxima- tions – why this is expected. Try to make this into a rigorous argument that applies to Rnfor any dimension n ≥ 1.
1.3 Curvature of plane and space curves
In a very general sense, differentiability makes precise the intuitive idea of being approximable by a linear object, think of tangent lines and planes, or more generally by linear functions and maps. Curvature then may loosely be thought of as a quantification how far the object is from locally being linear. Curvature is the central concept of differential geometry. In the case of graphs of functions y = f (x) all calculus students learn that the second derivative is somehow related to how much the graph curves – but it is important to fully understand that, and why, the second derivatives does not represent curvature.
Exercise 1.8 Consider the graph of f 0 (x) = x^2 for 0 ≤ x ≤ b. For small angles θ and small values of b > 0 the image of the graph under a rotation by an angle θ about the origin is again the graph of a function fθ(x). Find an explicit formula for fθ and show that its second derivative is not constant equal to f 0 ′′ ≡ 2. Suggestion: Use (x, y) to denote points on the original curve y = x^2 and let (ξ, η) denote points on the rotated curve. Express x and y in terms of ξ and η (compare exercise 1.7), substitute into y = x^2 and solve for η in terms of ξ. Finally calculate d
(^2) η dξ^2. Compare the associated MAPLE worksheet.
There are two aspects of the second derivative that do not make it suitable for immediate use to denote a notion of curvature: First the derivative in the exercise is taken with respect to the first coordinate x, as opposed to the intrinsic arc-length parameter. Secondly, the slopes are not the same as the direction of the curve – the tan in y′^ = tan α distorts the description.
In the following T, N, σ, κ,... are correct function names. However, for emphasis only, we often will write T (s), N (s), κ(s),... etc. This will also help distinguish from these from the compositions T ◦ ψ, κ ◦ ψ,... which with common abuse of notation, often are written as T (t), κ(t),.. .. One might even want to instead consider the functions (in our notation) T ◦ sigma−^1 , κ ◦ σ−^1 ,... whose domain are points in the image of the curve. However, for the purposes of differentiation etc., is is much easier to consider T, N, κ,... as functions defined on the parameter interval J.
Thus we first consider smooth curves σ : I 7 → R^2 and σ : I 7 → R^3 that are parameterized by arc- length. This implies that ‖σ′(s)‖ ≡ 1, i.e. the velocity is a unit tangent vector to the curve at
When given an initial reference frame R(0) consisting of three orthogonal unit vectors T (0), N (0) and B(0) together with two sufficiently regular functions κ(s) and τ (s) this system of differential equations uniquely determines R(s) for all times s, considering R as a curve in SO(3). Continuing further, if in addition an initial point σ(0) ∈ R^3 has been specified, then the Frenet Serret formulas together with the differential equation σ′^ = T (s) uniquely determine a curve in R^3. It is straightforward to verify that the curvature and torsion of this curve agree with the data provided to the differential equation. A most important corollary of this study is that the curvature and torsion completely determine a smooth curve up to translation (as determined by σ(0)) and rotation (as determined by R(0)).
Exercise 1.9 Explain how this gives as a corollary that curvature and torsion are invariant under translation and rotation (compare also exercise 1.7).
Exercise 1.10 Explore how complicated a brute force linear algebra calculation is (similar to exercise 1.7) that directly shows that curvature and torsion are invariant under rotations and translation. (It may be appropriate to use a computer algebra system for part of this work.)
Exercise 1.11 Show that if the curvature κ ≡ 0 of a plane curve vanishes identically, then the curve is a straight line. Is the same true for a curve in 3-space? Explain!
Exercise 1.12 Show that if the curvature κ ≡ c 6 = 0 of a plane curve is constant, then the curve is a circle with radius 1 /c. Is the same true for a curve in 3-space? Explain! (Remark: Feel free to consult the literature for elegant arguments – a direct brute-force approach quickly can get very messy!)
Exercise 1.13 Show that if the torsion τ ≡ 0 of a space curve vanishes identically, then the curve lies in a plane.
The Frenet-Serret formulas provide a most beautiful and comprehensive geometric description of the curves in 3-space. They appear to intrinsically rely on working with parameterizations by arc-length, yet for most curves explicit closed-form formulas for parameterizations by arc-length are beyond reach. However, note that all these formulas only involve derivatives of the curve σ. Consequently, there is no need to ever explicitly calculate the arc-length. All that is needed is the integrand of the formula (3) – the chain-rule does the rest. Consider a smooth curve γ : I = [a, b] 7 → Rn. Define ψ(t) =
∫ (^) t a
< γ′(τ ), γ′(τ ) > dτ. As usual, assume that ‖γ′(t)‖ > 0 for all t ∈ I. Then the curve σ : J = [0, L(γ)] 7 → Rn^ defined by σ = γ ◦ ψ−^1 is the reparameterization of γ by arc-length. Differentiating γ = σ ◦ ψ, the chain rule relates the velocities γ′^ = (σ′^ ◦ ψ) ψ′^ = ‖γ‖ T ◦ ψ – i.e. for each t ∈ I, the velocity vector v(t) = γ′(t) points in the same direction as T (ψ(t)) but it generally has non-unit magnitude (or “speed”) ‖γ′(t)‖. In practical calculations one typically first obtains γ′, then ‖γ′‖ and T. The key to avoiding excessively unpleasant calculations is to never differentiate normalized expressions such as T or N , but rather first take suitable cross- and dot-product of derivatives of γ. The next step is to note that γ′′^ = (σ′′^ ◦ψ) (ψ′)^2 +(σ′^ ◦ψ) ψ′′ implies that for every t ∈ I the vector γ′′(t) lies in the plane spanned by T (ψ(t)) and N (ψ(t)). (This plane is called the osculating plane.) In practical calculations in 3-space one calculates γ′′, then calculates B by normalizing the cross-product γ′^ × γ′′. Only afterwards(!) one calculates
N = B × T. Returning to the acceleration γ′′, the magnitudes of its tangential and normal components are easily calculated as
a‖ = (T ◦ ψ) · γ′′^ = γ′^ · γ′′ ‖γ′‖
and a⊥ = (N ◦ ψ) · γ′′^ = ±
‖γ′′‖^2 − a^2 ‖ (6)
The curvature κ (and radius of curvature % = (^1) κ ) and the torsion may be obtained in various ways. Typical formulas suitable for practical calculations (for space curves) are
κ ◦ ψ = | a⊥| ‖γ′‖^2
‖γ′^ × γ′′‖ ‖γ′‖^3
and τ ◦ ψ = (γ′^ × γ′′) · γ′′′ ‖γ′^ × γ′′‖^2
Exercise 1.14 Derive the formulas presented above for κ ◦ ψ and for τ ◦ ψ from the definitions of κ and τ in terms of the Frenet formulas.
Exercise 1.15 Consider a smooth planar curve γ : I 7 → R^2 , not necessarily parameterized by arclength. Devise a practical strategy to calculate T, N, κ with minimal effort. (Note, that from T one easily obtains N by interchanging the components and changing the sign of one of the components. Which one? Why?)
Exercise 1.16 For curves in the plane given as graphs of functions, i.e. γ(t) = (t, f (t)) or casually y = f (x), derive the usual formula κ = y′′/(1 + (y′)^2 )^3 /^2 for the curvature. Note that technically the above stands for κ ◦ γ 1 ◦ ψ = γ 2 ′′ /(1 + (γ′ 2 )^2 )^3 /^2.
Exercise 1.17 Verify that the curve γ : R 7 → R^2 defined by γ(t) = (exp(− 1 /t^2 ), 0) if t > 0 , γ(0) = (0, 0) and γ(t) = (0, exp(− 1 /t^2 ), 0) if t < 0 is infinitely many times continuously dif- ferentiable on the whole real line. Describe the (T (s), N (s)-frame for this curve (this is very easy), with special attention to what happens when t = 0.
Project 1.18 Write a MAPLE procedure that takes as input a space curve, i.e. algebraic ex- pressions for x(t), y(t) and z(t) (and possibly other parameters such as the domain I), and which gives as output an animation of the Frenet frame along the curve. As test curves consider a helix γ(t) = (cos t, sin t, t) and the (2,3)-torus-knot γ = u−^1 ◦ where(t) = (2t, 3 t) for t ∈ [0, 2 π] and u−^1 (θ, φ) = ((R + r sin φ) cos θ, (R + r sin φ) sin θ, r cos φ), e.g. for R = 5 and r = 2.
Project 1.19 Consider a family of closed curves γ(s, t) that are parameterized by arc-length s and that evolve with time t according to e.g. the heat-equation ∂ 2 ∂s^2 κ(s, t)^ −^
∂ ∂t κ(s, t) = 0.^ In- tuitively, this is the easiest model that describes how loops may try to straighten out under the influence of tension. For simplicity, start with initial curvature functions κ(s, 0) that are expressed as (finite) Fourier polynomials κ(s, 0) = a 0 +
∑N
j=1(aj^ cos^ jt^ +^ bj^ sin^ jt). This allows one to explicitly write out the solutions κ(s, t) = a 0 +
∑N
j=1 e −j^2 t(aj cos jt + bj sin jt) of the heat equation.
First find conditions on the Fourier coefficients that assure that the associated curve is closed. Then integrate the two-dimensional analogue of the Frenet-Serret formulas to obtain the associ- ated curve σ(t, s). Animate the images of the curves.
An exploratory worksheet that addresses this project is available from the WWW-site http://math.asu.edu/˜kawski/MAPLE/MAPLE.html. However, it has at least a cosmetic flaw as it arbitrarily fixes σ(0, t) and (^) ∂s∂ σ(0, t) – a nicer solution would include a more physical solution,
2 Manifolds
2.1 Introduction
We want to think of manifolds as abstractions and generalizations of the intuitive notions of curves and surfaces. This subsection reviews a few key ideas, purposes and examples. The next subsection provides a few fundamental topological notions to prepare for a precise definition of manifolds, first in the topological category, and then in the differential category.
The upcoming definition will characterize a manifold as a space which is such that every point in it has a neighbourhood that is homeomorphic to an open subset of a Euclidean space Rn. In particular, we shall not allow for edges and boundaries to avoid the associated technical complications. Next, we will equip manifolds with differentiable structures that allow for notions such as dynamical systems evolving on the manifolds, and for generalized notions of curvature. Typical objectives are to analyze the effects of curvature on the global topological structure or on the behaviour of dynamical systems. A need to integrate over (subsets of) manifolds arises naturally. A major role of local coordinate charts is to transfer these differential (and integral) concepts back into familiar Euclidean space where standard techniques may be employed for calculations.
Throughout we will emphasize geometric points of view – as a simple example what we don’t want think of the two dimensional sphere S^2 as (the union of) the graph(s) of two functions z = ±
x^2 + y^2. This rather arbitrary preferential treatment of z versus x and y begins to hide the full symmetry of the sphere under a group of rotations and reflections.
Before proceeding to technical descriptions let us take a brief look at some typical examples that should be included in our notion of manifold. Curves and surfaces, especially the Euclidean spaces Rn, and (open) subsets of Euclidean spaces should be manifolds. However, we may impose conditions so as to avoid e.g. self-intersections, boundaries, and, in the category of differentiable manifolds, cusps, corners and the like. The characterization of the two dimensional sphere S^2 ⊆ R^3 as the set of all (x, y, z) ∈ R^3 that satisfy x^2 + y^2 + z^2 = 1, invites a natural generalization to higher dimensional analogues of surfaces as subsets of R that may be characterized by (sets of) equations Fk(x 1 , x 2 ,... xn) = 0 (k = 1,... p). To avoid cusps and corners one usually imposes a condition that the gradient (or a higher dimensional analogue) does not vanish. As a special case, this description immediately opens the door to objects such as the group of special orthogonal n × n matrices SO(n). The defining equation AT^ A = In×n is of the same form as the equation of the sphere given above. What makes these matrix manifolds particularly interesting is their natural group structure – there is a natural notion of multiplying points on the generalized surface – this is the starting point for Lie groups. A different way that many manifolds of interest are obtained is by taking quotients. In the most simple case the circle S^1 arises as a quotient of R by Z. Intuitively, for any periodic function f with period p > 0, i.e. f (x + p) = f (x) for all x ∈ R, one may consider as its natural domain any interval [a, a + p] with endpoints identified. More abstractly, consider the equivalence relation ∼ defined on R by x ∼ y ⇐⇒ (x − y)/p ∈ Z. Then each point on the circle Θ represents an equivalence class [Θ] = {Θ + k p : k ∈ Z}. In an analogous way, the torus arises naturally (e.g. very commonly in dynamical systems) as the quotient of the plane R^2 by Z^2. One commonly visualizes the torus as the unit square [0, 1] × [0, 1] with opposing edges identified.
Picture: squares with identified edges, cylinder, torus, Klein bottle
If one starts with the same square, but identifies one (or two) sets of opposing edges with orientation reversed one arrives at the Klein bottle and at the projective plane. Neither one of these can be visualized in the usual way as a surface in R^3 , but apparently each shares many properties with the torus due to their analogous construction. More abstractly, projective spaces arise when considering the spaces of all (straight) lines in Rn^ that pass through the origin. Before looking at this more closely, recall the simple case of considering the space of all (semi-infinite, open) rays emanating from the origin. Each of these rays may be naturally identified with the point on the unit sphere (unit circle) through which it passes. Thus we may think of the spheres Sn−^1 as arising from Rn^ \ { 0 } as quotients under the equivalence relation x ∼ y ⇐⇒ if there exists λ ∈ R, λ > 0 such that x = λy. In a practical sense this is closely related to considering only the angle(a) θ (or (θ, φ)) when working with polar (or spherical coordinates). In analogy, if one discards the requirement λ > 0 in the preceding definition of equivalence, then the equivalence classes are the lines through the origin. (More precisely, since we started with Rn^ \ { 0 }, the origin is removed from each line, or the line really consists of two rays.) One may visualize the resulting quotient space as the space of pairs of opposite points on the sphere Sn−^1 , or as a semi-sphere with two halves of the equator (which is a sphere Sn−^2 by itself) identified, or glued together with careful attention to the orientation of each piece. From these projective spaces it is only a small step to Grassmannian manifolds which may be thought of as spaces of m-dimensional (hyper-)planes in n-dimensional Euclidean space. A typical application where these appear naturally is in the classification of linear control systems x˙ = Ax + Bu, with state x ∈ Rnand control u ∈ Rm. Here one considers two systems equivalent if one may be transformed into the other by coordinates changes, in state ˜x = Rx and control space ˜u = Su, and/or under feedback transformations u˜ = u + Kx,...
All the above examples clearly have (preserve) some additional structure beyond just being sets of points. In order to be able to work with concepts such as continuity and notions of derivatives one intuitively needs some notion of distance. Indeed, while one can start with even more general topological spaces, in the finite dimensional setting very little is lost if one requires that the set is equipped with at least some a-priori notion of distance. However, this basic notion of distance will primarily be used only as a foundation for e.g. continuity, and should not be confused with the Riemannian metrics that we will study later, and which have a deeply connected with curvature.
The notions of open and closed do not require an underlying metric structure. The following axioms allow for a generalization to spaces without a metric:
Definition 2.3 A topology on a set X is a collection T of subsets of X that satisfies
(i) ∅ ∈ T and X ∈ T ,
(ii) T is closed under (arbitrary) unions, i.e., if {Oα : α ∈ A} ⊆ T then
α∈A Oα^ ∈ T^ , and (iii) T is closed under finite intersections, i.e., if Ok ∈ T , k = 1, 2 ,... n, then
⋂n k=1 Ok^ ∈ T^.
A subset O ⊂ X is called open if O ∈ T. A subset F ⊂ X is called closed if X \ F ∈ T. A topological space is a pair (X, T ) where X is a set and T is a topology on X.
Note that an infinite intersection of open sets is not required to be open. The standard example is the real line with the usual topology and Ok = (− (^1) k , (^) k^1 ). Clearly
k=1 Ok^ =^ {^0 }^ which is not open (in the standard topology). One commonly uses the term (open) neighborhood of p ∈ X for an open set which contains p. While technically a topological space is a pair (X, T ), one often refers to X alone as a topological space. In such cases it is usually understood from the context which topology T on X is meant. Commonly one specifies a topology by describing a smaller set of basic open sets.
Definition 2.4 If X is a set, a collection B of subsets of X is a basis for a topology on X if it satisfies
(i) For every x ∈ X there exists B ∈ B such that x ∈ B.
(ii) For all B 1 , B 2 ∈ B and all x ∈ B 1 ∩ B 2 there exists B 3 ∈ B such that x ∈ B 3 ⊆ B 1 ∩ B 2.
The topology generated by B consists of all subsets O ⊆ X such that for every x ∈ O, there exists a set B ∈ B such that x ∈ B ⊆ O.
Exercise 2.6 Show that if Y ⊆ X is a subset of a topological space (X, T ) then the collection T ′^ = {O ∩ Y : O ∈ T } defines a topology on Y. This topology is called the subspace topology.
Exercise 2.7 Suppose that (X, d) is a metric space. Show that the collection of open sets (in the sense of open in a metric space) defines a topology on X. (This topology is called the metric topology on (X, d).)
Definition 2.5 A topological space (X, T is first countable if at every x ∈ X it has a countable basis, that is, for every x ∈ X there exists a countable collection {Bkx ∈ T : k ∈ Z+} such that for every O ∈ T , if x ∈ O then there exists k ∈ Z such x ∈ Bxk ⊆ O. A topological space (X, T is second countable if it has a countable basis. A topological space (X, T is separable if there exists a countable dense subset Y ⊆ X, i.e. a subset Y ⊆ X such that for every x ∈ X and every O ∈ T , if x ∈ O then O ∩ Y 6 = ∅.
Example 2.1 Every metric space is first countable. Every separable first countable space is second countable. Every Euclidean space Rn^ is second countable.
Exercise 2.8 Prove the assertions made in example 2.1.
Definition 2.6 Suppose X and Y (or, more precisely (X, TX ) and (Y, TY )) are topological spaces. A map f : X 7 → Y is called continuous if for every open set O ⊆ Y the preimage f −^1 (O) ⊆ X is open (i.e. O ∈ TY =⇒ f −^1 (O) ∈ TX ). (This is equivalent to f −^1 (F ) ⊆ X closed for every F ⊆ Y closed.) A map f : X 7 → Y is called open if for every open set O ⊆ X the image f (O) ⊆ Y is open. A map f : X 7 → Y is called closed if for every closed set F ⊆ X the image f (F ) ⊆ Y is closed.
In the case that TX and TY are the metric topologies associated with metrics dX and dY on X and Y , respectively, this notion of continuity agrees with the standard ε-δ characterization of continuity. A function f : X 7 → Y is continuous (as defined above) if and only if for every p ∈ X and for every ε > 0 there is a δ > 0 such that for all q ∈ X if dX (p, q) < δ then dY (f (p), f (q)) < δ. Practically the notion of continuity captures the concept that small changes in the input of a function cause only small changes in the output.
Exercise 2.9 Consider the set R of real numbers with the usual topology T 2 , with the indiscrete topology T 1 = {∅, R}, and with the discrete topology T 3 in which every subset of R is open. For each pair (i, j) with i, j = 1, 2 , 3 describe the set of continuous functions from (R, Ti) to (R, Tj ). (Make a 3 × 3 table.) In particular, for which pairs is the identity function id : x 7 → x continuous? For which pair(s) are (only) the constant functions continuous, and for which pair(s) are all functions continuous?
Exercise 2.10 Verify the assertion that in metric spaces the standard ε-δ characterization of continuity agrees with the definition given above.
One of the most common uses of connectedness is the argument that if a function f : X 7 → R is continuous and locally constant then it is constant provided the domain X is connected. Here, locally constant means that every p ∈ X has an open neighbourhood U (containing p) such that the restriction of f to U is constant. To clarify this argument, consider the function f : R \ { 0 } 7 → R defined by f : x 7 → 0 if x < 0 and f : x 7 → 1 if x > 0. Clearly the derivative f ′^ ≡ 0 vanishes identically, but f is not constant. Of course, the key is that the domain is not connected. Consequently, the vanishing of the derivative only assures that f is locally constant. It does not assure that f is constant.
Arguably the most important topological concept for us is compactness. One may think of it as an outgrowth of the desire to generalize, or to get to the underlying foundation of the important theorem that every continuous function f : [a, b] 7 → R defined on a closed bounded interval attains its minimum and its maximum, i.e. there exist points x 1 , x 2 ∈ [a, b] such that for all x ∈ [a, b], , f (x 1 ) ≤ f (x) ≤ f (x 2 ). It is well-known from calculus that this assertion may fail if either of the closedness or boundedness hypotheses is omitted. The closedness requirement naturally generalizes to general topological spaces, but the boundedness does not. For example, if Rn^ is equipped with the bounded metric d¯ = d 2 /(1 + d 2 ) (where d 2 is the standard Euclidean metric), then (Rn, d¯) still has the same topology as (Rn, d 2 ) yet while K = Rn^ is closed and bounded in (Rn, d¯), it is not in (Rn, d 2 ). Many different generalizations have been proposed to generalize the basic idea of “closed and bounded” which is so useful in Rn(with its usual metric). Any introductory course in point-set topology will discuss such different notions of compactness. It was not until quite late into the 20th^ century that the following notion finally crystallized, and it became clear that it captures the fundamental features of the desired properties.
Definition 2.10 A subset K ⊆ X of a topological space X is called compact if every open cover of⋃ K has a finite subcover, i.e. if {Oα ⊆ X : α ∈ A} is a collection of open sets such that K ⊆
α∈A Oα^ then there exists a finite subcollection^ {Oαj :^ j^ = 1,^2 ,... n}^ such that^ K^ ⊆^
⋃n j=1 Oαj
The Heine-Borel theorem asserts that every bounded closed interval in R is compact. Its proof may be found in any advanced calculus text.
Exercise 2.17 Prove that if f : X 7 → Y is continuous and K ⊆ X is compact then the image f (K) ⊆ Y is compact. In the case of Y = R this implies that there exist points p, q ∈ X at which f attains its global minimum and global maximum, i.e. such that f (p) ≤ f (x) ≤ f (q) for all x ∈ X.
Definition 2.11 A sequence {ak}k∈N ⊆ X in a topological space X is said to converge if there exists x¯ ∈ X such that for every open set O ⊆ K containing x¯ there exist a finite natural number N such that an ∈ O for all n > N.
Exercise 2.18 Suppose {xk}k∈N is an infinite sequence with values in a compact topological space K. Show that {xk}k∈N has an accumulation point x¯ ∈ K. If, in addition, K is first countable then there exists a converging subsequence {xkj }j∈N.
Finally we mention a few separation axioms which on occasion are used as essential hypotheses in differential geometry.
Definition 2.
- A topological space X is called a Hausdorff space if for every pair of distinct points p, q ∈ X there exist disjoint open sets U and V such that p ∈ U and q ∈ V.
- A Hausdorff space X is called completely regular if one-point sets are closed in X and if for every point p ∈ X and every closed set F ⊆ X not containing p there exists a continuous function f : X 7 → R such that f (p) = 0 and f (x) = 1 for every x ∈ F.
- A Hausdorff space X is called normal if one-point sets are closed in X and if for every pair of disjoint closed sets F 1 , F 2 ⊆ X there exists disjoint open sets U and V such that F 1 ⊆ U and F 2 ⊆ V.
These notions become useful when patching together local results, e.g. obtained in one co- ordinate chart at a time. This will be made precise when discussing partitions of unity in a subsequent section. For the sake of completeness we here also give the following two technical definitions:
Definition 2.13 A map f : X 7 → Y is called proper if for every compact set K ⊆ Y the preimage f −^1 (K) ⊆ X is compact.
Definition 2.14 A subset S ⊆ X of a Hausdorff space X is called paracompact if every open cover of S has a locally finite open refinement. This means that if Uα, α ∈ A are open sets such that S ⊆
α∈A Uα^ then there exist a collection of open sets^ Vβ^ , β^ ∈^ B^ such that
- For every β ∈ B there exists an α ∈ A such that Vβ ⊆ Uα,
- S ⊆
β∈A Vβ^ , and
- every p ∈ S has an open neighbourhood W which intersects only a finite number of the sets Vβ , β ∈ B.
Paracompactness is very close to metrizability, (indeed, metrizability is equivalent to paracom- pactness and local metrizability). Thus many authors use paracompactness as a basic require- ment when defining manifolds.
- The open ball Bp(r) = {x ∈ Rn^ : ‖x−p‖ < r} of radius r about p ∈ Rn^ is an n-dimensional manifold with a single chart given by U = Bp(r) and u(x) = (x − p)/(r − ‖x − p‖).
Exercise 2.19 Verify that the inverse is given by u−^1 (y) = p + ry/(1 + ‖y‖).
- Every open subset of an n-dimensional manifold is itself an n-dimensional manifold.
- Identify the space Mm,n(R) of m-by-n matrices with real entries with the space Rmn. E.g. in the 2 × 2-case simply identify
u :
a b c d
a b c d
Note that with this identification Mm,n(R) inherits a natural metric structure. Clearly this shows that the entire space Mm,n(R) is an nm-dimensional manifold. Of more interest are various subspaces of Mmn(R). Typical examples are the general linear group GL(n, R) = {A ∈ Mn,n(R) : det A 6 = 0} and its subset of orientation preserving nonsingular matrices GL+(n, R) = {A ∈ Mn,n(R) : det A > 0 }. Both are n^2 -dimensional manifolds. The argument uses that det is a polynomial function in the entries of the matrix, and hence it is continuous. Consequently, the preimages det−^1 (R \ { 0 }) and det−^1 (0, ∞)) of open sets are open subsets of Mn,n(R). Further examples are the special linear groups SL(n, R) = {A ∈ Mn,n(R) : det A = 1}, the orthogonal groups O(n) = {A ∈ Mn,n(R) : AT^ A = I}, and the special orthogonal groups SO(n) = O(n) ∩ SL(n, R). Unlike the previous examples – which are open subsets – these manifolds are defined by closed conditions (i.e. “=” as opposed to “ 6 =”, “<” or “>”). We return to these examples later when tools from differential calculus on manifolds will make it easy to establish when such subsets defined by closed conditions give rise to manifolds.
- The m-sphere Sm(r) = {x ∈ Rm+1^ : ‖x‖ = r} is a m-manifold. The case m = 0 is special with S^0 (r) = {−r, r} ⊆ R consisting of only two points, and thus being disconnected. The 1-sphere is a circle, and one needs at least two charts, e.g. U = S^1 (r) \ {(−r, 0)} and V = S^1 (r) \ {(r, 0)}. Use u(x, y) = atan2(x, y) taking values in (−π, π) and v(x, y) = atan2(x, y) taking values in (0, 2 π). Note that the local coordinates agree essentially with the angle of polar coordinates.
Exercise 2.20 Construct a collection of charts for the 2 -sphere S^2 (r) by explicitly adapt- ing the spherical coordinates u(x, y, z) = (θ(x, y, z), φ(x, y, z)) ∈ (0, 2 π) × (0, π) to different subsets resulting from different “cuts” What are the minimal number of cuts, and the minimal number of charts needed to cover S^2?
A different set of coordinates, that is particularly useful for higher dimensional spheres is based on stereographic projections: Consider the two subsets U± = Sm(r){(0, 0 ,... , ±r)} and define the maps u± : U± 7 → Rm^ by
u±(x) = 2 r r ∓ xm+
(x 1 ,... xm) (9)
Graphically (u±(x), ∓r) ∈ Rm+1) is the point where the hyperplane xm+1 = ∓r intersects the line that passes through the point x ∈ Sm(r) and through (0, 0 ,... , ±r).
Exercise 2.21 For the inverse maps u− ±^1 : Rm^7 → U± ⊆ Sm(r), derive the formulae
x = u− ±^1 (y) =
2 r 1 +
∥ 2 yr
∥^2
( (^) y 1 2 r
2 r 1 +
∥ 2 yr
∥^2
( (^) y m 2 r
, ∓ r ·
∥ 2 yr
∥^2
∥ 2 yr
∥^2
Use this to obtain explicit formulae for the “transition maps” u∓ ◦ u− ±^1 : Rm^7 → Rm. What do these maps do graphically – e.g. which sets do they leave fixed? What are the images of (special) lines and circles?
- If M m^ and N n^ are m- and n−dimensional manifolds, respectively, then the Cartesian product M × N is an (m + n)-dimensional manifold: Suppose (p, q) ∈ M × N and (u, U ) and (v, V ) are coordinate charts about p and q, then (u × v, U × V ) is a coordinate chart about (p, q) where (u, v)(a, b) = (u(a), v(b)) ∈ Rm^ × Rn. A typical example uses that the circle S^1 = {x ∈ R^2 : ‖x‖ = 1} is a manifold to establish that the torus T 2 = S^1 × S^1 is a 2-dimensional manifold.
- We briefly return to the real projective spaces, now illustrating coordinate charts. On Rm+1^ \ { 0 } define the equivalence relation x ∼ y if x = λy for some λ ∈ R. The m- dimensional real projective space is defined as the quotient IPm^ =
Rm+1^ \ { 0 }
/∼, i.e. a point [x] ∈ IPm^ is the equivalence class [x] = {y ∈ Rm+1^ \ { 0 } : y ∼ x}. Intuitively think of IPm^ as the space of all lines in Rm+1^ that pass through the origin, or as the m-sphere Sm^ with antipodal points x and −x identified. More graphically, one may obtain IP^2 by sewing a disk to the (only one!) edge of a M¨obius strip. For j = 1,... m consider the sets Uj = {[x] ∈ IPm^ : xj 6 = 0} and coordinates maps (homogeneous coordinates)
uj ([x]) =
x 1 xj
xj− 1 xj
xj+ xj
xm+ xj
It is straightforward to verify that the value of uj ([x]) does not depend on the choice of the representative x ∈ [x]. The inverse is given by u− j 1 (y 1 ,... ym) = [y 1 ,... , yi− 1 , 1 , yi+1,... ym].
A complete discussion of these coordinate maps (they are supposed to be homeomorphisms) is straightforward but technical, in terms of the quotient topology. For an introductory discussion of quotient maps, and quotient topologies see e.g. Munkres (Topology, a first course”, p.134). A main issue is to assure that the quotient topology is not pathological. Just as a reference, a surjective map f : X 7 → Y is called a quotient map if O ⊂ Y is open if and only if f −^1 (O) ⊆ X is open. For any map f : X 7 → A from a topological space X to a set A there is exactly one topology on A, called the quotient topology, such that f is a quotient map. In the case that A is a set of equivalence classes on X, A with this topology is called a quotient space of X.
- Two-dimensional surfaces in R^3 , or more generally m-dimensional hypersurfaces in Rm+ are some of familiar manifolds. Clearly every graph {(x, f (x)) : x ∈ Rm} ⊆ Rm+1^ of any continuous function f : Rm^7 → R is a manifold with a single chart u : (x, f (x)) 7 → x ∈ Rm. More interesting are hypersurfaces that arise as preimages (“zero-sets”) M = F −^1 ({ 0 }) ={x ∈ Rm+1^ : F (x) = 0} of functions F : Rm+1^7 → R, or that are given by parameterizations F : Rm^7 → Rm+1^ and M = F (Rm).
