## Lecture 2: Smooth Manifolds and Scalar Fields

We now generalize the ideas discussed in Lecture 1.

 Definition 2.1 An open cover of M Es is a collection {Ua} of open sets in M such that M = aUa.

Examples

(a) Es can be covered by open balls.

(b) Es can be covered by the single (open) set Es.

(c) The unit sphere in Es can be covered by the collection {U1, U2} where

U1 = {(y1, y2, y3) | y3 > -1/2}
U2 = {(y1, y2, y3) | y3 < 1/2}.

 Definition 2.2 A subset M of Es is called an n-dimensional smooth manifold if we are given a collection {Ua; xa1, xa2, . . ., xan} where: (a) The Ua form an open cover of M. (b) Each xar is a smooth (what does that mean?) real-valued function defined on U (that is, xar: UaE1), called the th coordinate, such that the map x: UaEn given by x(u) = (xa1(u), xa2(u), . . . , xan(u)) is one-to-one. (That is, to each point in Ua, we are assigned a unique set of n coordinates.) The tuple (Ua; xa1, xa2, . . ., xan) is called a local chart of M. The collection of all charts is called a smooth atlas of M. Further, Ua is called a coordinate neighborhood. (c) If (U, xi), and (V, j) are two local charts of M, and if UV , then we can write xi = xi(j) with inverse k = k(xl) for each i and k, where all functions in sight are smooth. These functions are called the change-of-coordinates transformations.

By the way, we call the "big" space Es in which the manifold M is embedded the ambient space.

Notes
1. Always think of the xi as the local coordinates (or parameters) of the manifold. We can paramaterize each of the open sets U by using the inverse function x-1 of x, which assigns to each point in some neighborhood of En a corresponding point in the manifold. Let me see an example.
2. Condition (c) implies that

det ixj
0, and

det xij
0,

since the associated matrices must be invertible.
3. The ambient space need not be present in the general theory of manifolds; that is, it is possible to define a smooth manifold M without any reference to an ambient space at all -- see any text on differential topology or differential geometry.
4. More terminology: We shall sometimes refer to the xi as the local coordinates, and to the yj as the ambient coordinates. Thus, a point in an n-dimensional manifold M in Es has n local coordinates, but s ambient coordinates.

Examples 2.3
(a) En is an n-dimensional manifold, with the single identity chart defined by

xi(y1, . . . , yn) = yi.
(b) S1, the unit circle, with the exponential map, is a 1-dimensional manifold. Here is a possible structure:with two charts as show in in the following figure.

One has

x: S1-{(1, 0)} E1 : S1-{(-1, 0)} E1,
with 0 < x, < 2, and the change-of-coordinate maps are given by

=
 +x if x < -x if x >
(See the figure for the two cases.)
and
x =
 + if < - if >
,

Notice the symmetry between x and . Also notice that these change-of-coordinate functions are only defined when 0, . Further,

/x = x/ = 1.

Note that, in terms of complex numbers, we can write, for a point p = eiz S1,

x = arg(z), = arg(-z).

(c) Generalized Polar Coordinates Let us take M = Sn, the unit n-sphere,

Sn = {(y1, y2, ... , yn, yn+1) En+1 | iyi2 = 1},

with coordinates (x1, x2, . . . , xn) with

0 < x1, x2, . . . , xn-1 < ,   and
0 < xn < 2,
given by

y1 = cos x1
y2 = sin x1 cos x2
y3 = sin x1 sin x2 cos x3
...
yn-1 = sin x1 sin x2 sin x3 sin x4 ... cos xn-1
yn = sin x1 sin x2 sin x3 sin x4 ... sin xn-1 cos xn
yn+1 = sin x1 sin x2 sin x3 sin x4 ... sin xn-1 sin xn

In the homework, you will be asked to obtain the associated chart by solving for the xi. Note that if the sphere has radius r, then we can multiply all the above expressions by r, getting

y1 = r cos x1
y2 = r sin x1 cos x2
y3 = r sin x1 sin x2 cos x3
...
yn-1 = r sin x1 sin x2 sin x3 sin x4 ... cos xn-1
yn = r sin x1 sin x2 sin x3 sin x4 ... sin xn-1 cos xn
yn+1 = r sin x1 sin x2 sin x3 sin x4 ... sin xn-1 sin xn.

(d) The torus T = S1S1, with the following four charts:

x: (S1-{(1, 0)})(S1-{(1, 0)})E2, given by
x1((cos, sin), (cos, sin)) =
x2((cos, sin), (cos, sin)) = .

The remaining charts are defined similarly, and the change-of-coordinate maps are omitted.

(e) The cylinder (exercise)

(f) Sn, with (again) stereographic projection, is an n-manifold; the two charts are given as follows. Let P be the point (0, 0, . . , 0, 1) and let Q be the point (0, 0, . . . , 0, -1). Then define two charts (Sn-P, xi) and (Sn-Q, i) as follows. (See the figure.)

If (y1, y2, . . . , yn, yn+1) is a point in Sn, let

x1= y11 - yn+1

1= y11 + yn+1
x2= y21 - yn+1

2= y21 + yn+1
. . .   . . .
xn= yn1 - yn+1

n= yn1 + yn+1

We can invert these maps (that is, solve for the global coordinates yi in terms of the local coordinates xi and i) as follows:

Let r2 = i xixi, and 2 = i ii. Then:

y1= 2x1r2 + 1

y1= 211+2
y2= 2x2r2 + 1

y2= 221+2
. . .   . . .
yn= 2xnr2 + 1

yn= 2n1+2
yn+1= r2 - 1r2 + 1

yn+1= 1 - 21+2

The change-of-coordinate maps are therefore:

x1=
 y11 - yn+1
=
21

1+2

1- 1 - 21 + 2
=  12
x2=  22
. . .
xn=  n2

This makes sense, since the maps are not defined when i = 0 for all i, corresponding to the north pole.

Note Since is the distance from i to the origin, this map is hyperbolic reflection in the unit circle;

xi = 1
 i

r = 1/

That is, project it to the circle, and invert the distance from the origin. This also gives the inverse relations, since we can write

i = 2xi = xi/r2.

In other words, we have the following transformation rules.

Change of Coordinate Transformations for Stereographic Projection

Let r2 = i xixi, and 2 = i ii.

Then   i=
xi

r2
;     i=
 i2
;     r=
 1

Note We can put all the coordinate functions xar: UaE1 together to get a single map

xa: UaWa En.

A more precise formulation of condition (c) in the definition of a manifold is then the following: each Wa is an open subset of En, each xa is invertible, and each composite

 Wa xa-1 En xb Wb

is a smooth function defined on an open subset.

We now want to discuss scalar and vector fields on manifolds, but how do we specify such things? First, a scalar field.

 Definition 2.4 A smooth scalar field on a smooth manifold M is just a smooth real-valued map : ME1. (In other words, it is a smooth function of the coordinates of M as a subset of Er.) Thus, associates to each point m of M a unique scalar (m). If U is a subset of M, then a smooth scalar field on U is smooth real-valued map : UE1. If U M, we sometimes call such a scalar field local.

If is a scalar field on M and x is a chart, then we can express as a smooth function of the associated parameters x1, x2, . . . , xn. If the chart is , we shall write for the function of the other parameters 1, 2, . . . , n. Note that we must have = at each point of the manifold (see the transformation rule below).

Examples 2.5 (a) Let M = En (with its usual structure) and let be any smooth real-valued function in the usual sense. Then, using the identity chart, we have = .

(b) Let M = S2, and define (y1, y2, y3) = y3.
Using stereographic projection, we find both and :

(x1, x2)= y3(x1, x2)=  r2 - 1r2 + 1
=  (x1)2 + (x2)2 - 1(x1)2 + (x2)2 + 1
(1, 2)= y3(1, 2)=  1 - 21 + 2
=  1 - (1)2 - (2)21 + (1)2 + (2)2

(c) Local Scalar Field The most obvious candidate for local fields are the coordinate functions themselves. If U is a coordinate neighborhood, and x = {xi} is a chart on U, then the maps xi are local scalar fields.

Sometimes, as in the above example, we may wish to specify a scalar field purely by specifying it in terms of its local parameters; that is, by specifying the various functions instead of the single function . The problem is, we can't just specify it any way we want, since it must give a value to each point in the manifold independently of local coordinates. That is, if a point p M has local coordinates (xj) with one chart and (h) with another, they must be related via the relationship

j = j(xh).

 Transformation Rule for Scalar Fields (j) = (xh).

Example 2.6 Look at Example 2.5(b) above. If you substituted i as a function of the xj, you would get (1, 2) = (x1, x2) (after some laborious albegra!).

Exercise Set 2

1. Give the paraboloid z = x2 + y2 the structure of a smooth manifold.

2. Find a smooth atlas of E2 consisting of three charts.

3. (a) Extend the method in Exercise 1 to show that the graph of any smooth function f: E2E1 can be given the structure of a smooth manifold.
(b) Generalize part (a) to the graph of a smooth function f: En E1.

4. Two atlases of the manifold M give the same smooth structure if their union is again a smooth atlas of M.
(a) Show that the smooth atlases (E1, f), and (E1, g), where f(x) = x and g(x) = x3 are incompatible.
(b) Find a third smooth atlas of E1 that is incompatible with both the atlases in part (a).

5. Consider the ellipsoid L E3 specified by

 x2a2
+ y2b2
+ z2c2
=1,

(a, b, c 0). Define f: LS2 by f(x, y, z) = (x/a, y/b. z/c).
(a) Verify that f is invertible (by finding its inverse).
(b) Use the map f, together with a smooth atlas of S2, to construct a smooth atlas of L.

6. Find the chart associated with the generalized spherical polar coordinates described in Example 2.3(c) by inverting the coordinates. How many additional charts are needed to get an atlas? Give an example.

7. Obtain the equations in Example 2.3(f).

Last Updated: january, 2002