A very brief introduction to differentiable manifolds

Tom Carter

http://cogs.csustan.edu/~tom/diff-manifolds

Santa Fe Institute
Complex Systems Summer School

June, 2001

Contents

• Differentiable manifolds can generally be thought of as a generalization of Rⁿ. They are mathematical objects equipped with smooth (local) coordinate systems. Much of physics can be thought of as having a natural home in differentiable manifolds. A particularly valuable aspect of differentiable manifolds is that unlike traditional flat (Euclidean) Rⁿ, they can have (intrinsic) curvature.

• We need a way to talk about ``nearness'' of points in a space, and continuity of functions. We can't (yet) talk about the ``distance'' between pairs of points or limits of sequences - we will use a more abstract approach. We start with:

  Def.: A topological space (X, T) is a set X together with a topology T on X. A topology on a set X is a collection of subsets of X (that is, T Ì P(X)) satisfying:

  1. If G₁, G₂ Î T, then G₁ ÇG₂ Î T.
  2. If {G_a | a Î J} is any collection of sets in T, then

     È a Î J  Ga Î T.

  3. Æ Î T, and X Î T.

• The sets G Î T are called open sets in X. A subset F Ì X whose complement is open is called a closed set in X.

• If A is any subset of a topological space X, then the interior of A, denoted by A^°, is the union of all open sets contained in A. The closure of A, denoted by [`A], is the intersection of all closed sets containing A.

• If x Î X, then a neighborhood of x is any subset A Ì X with x Î A^°.

• If (X, T) is a topological space, and A is a subset of X, then the induced or subspace topology T_A on A is given by

  TA = {G ÇA | G Î T}.

  It is easy to check that T_A actually is a topology on A. With this topology, A is called a subspace of X.

• Suppose X and Y are topological spaces, and f : X ® Y. Recall that if V Ì Y, we use the notation

  f-1(V) = {x Î X | f(x) Î V}.

  We then have the definition:

  Def.: A function f : X ® Y is called continuous if f^-1(G) is open in X for every open set G in Y.

• We can also define continuity at a point. Suppose f : X ® Y, x Î X, and y = f(x). We say that f is continuous at x if for every neighborhood V of y, there is a neighborhood U of x with f(U) Ì V. We then say that a function f is continuous if it is continuous at every x Î X.

• A homeomorphism from a topological space X to a topological space Y is a 1-1, onto, continuous function f : X ® Y whose inverse is also continous.

• A topological space is called separable if there is a countable collection of open sets such that every open set in T can be written as a union of members of the countable collection.

• A topological space X is called Hausdorff if for every x, y Î X with x ¹ y, there are neighborhoods U and V of x and y (respectively) with U ÇV = Æ.

• This is just the barest beginnings of Topology, but it should be enough to get us off the ground ...

1. Show that the intersection of a finite number of open sets is open. Give an example to show that the intersection of an infinite number of open sets may not be open.
2. How many distinct topologies are there on a set containing three elements?
3. Show that the interior of a set is open. Show that the closure of a set is closed. Show that A^° Ì A Ì [`A]. Show that it is possible for A^° to be empty even when A is not empty.
4. Show that if f : X ® Y is continuous, and F Ì Y is closed, then f^-1(F) is closed in X.
5. Show that a set can be both open and closed. Show that a set can be neither open nor closed.
6. Show that if f : X ® Y and g : Y ® Z are both continuous, then g °f : X ® Z is continuous.
7. Show that the two definitions of continuity are equivalent.
8. A subset D Ì X is called dense in X if [`D] = X. Show that it is possible to have a dense subset D with D^° = Æ.
9. Show that if D is dense in X, then for every open set G Ì X, we have G ÇD ¹ Æ. In particular, every neighborhood of every point in X contains points in D.
10. Show that in a Hausdorff space, every set consisting of a single point x (i.e., {x}) is a closed set.

• For any set X, there are two trivial topologies:

  Tc = {Æ, X}

  and

  Td = P(X).

  T_d is the topology in which each point (considered as a subset) is open (and hence, every subset is open). It is called the discrete topology. T_c is sometimes called the concrete topology.

• On R, there is the usual topology. We start with open intervals (a, b) = {x | a < x < b}. An open set is then any set which is a union of open intervals.

• On Rⁿ, there is the usual topology. One way to get this is to begin with the open balls with center a and radius r, where a Î Rⁿ can be any point in Rⁿ, and r is any positive real number:

  Bn(a, r) = {x Î Rn | |x - a| < r}.

  An open set is then any set which is a union of open balls.

1. Check that each of the examples actually is a topological space.

2. For k < n, we can consider R^k to be a subset of Rⁿ. Show that the inherited subspace topology is the same as the usual topology.

3. Show that Rⁿ with the usual topology is separable and Hausdorff.

• Suppose M is a topological space, U is an open subset of M, and m: U ® Rⁿ. Suppose further that m(U) is an open subset of Rⁿ, and that m is a homeomorphism between U and m(U). We call m a local coordinate system of dimension n on U.

  For each point m Î U, we then have that m(m) = (m₁(m), ¼, m_n(m)), the coordinates of m with respect to m.

• Now suppose that we have another open subset V of M, and n is a local coordinate system on V. We say that m and n are C^¥ compatible if the composite functions m°n^-1 and n°m^-1 are C^¥ functions on m(U) Çn(V). Remember that a function on Rⁿ is C^¥ if it is continuous, and all its partial derivatives are also continuous.

• A topological manifold of dimension n is a separable Hausdorff space M such that every point in M is in the domain of a local coordinate system of dimension n. These spaces are sometimes called locally Euclidean spaces.

• A C^¥ differentiable structure on a topological manifold M is a collection F of local coordinate systems on M such that:
  1. The union of the domains of the local coordinate systems is all of M.
  2. If m₁ and m₂ are in F, then m₁ and m₂ are C^¥ compatible.
  3. F is maximal with respect to 2. That is, if n is C^¥ compatible with all m Î F, then n Î F.

• A C^¥ differentiable manifold of dimension n is a topological manifold M of dimension n, together with a C^¥ differentiable structure F on M.

  Notes:

  1. It is possible for a topological manifold to have more than one distinct differentiable structures.
  2. In this discussion, we have limited ourselves to C^¥ differentiable structures. With somewhat more work, we could define C^k structures for k < ¥.
  3. We have limited the domains of our local coordinate systems to be open subsets of M. This means that the usual spherical and cylindrical coordinate systems on R³ do not count as local coordinate systems by our definition.
  4. With somewhat more work, we could define differentiable manifolds with boundaries.
  5. We have limited ourselves to manifolds of finite dimension. With somewhat more work, we could define infinite dimensional differentiable manifolds.