Examples of Differentiable Manifolds

Do Carmo’s Riemannian Geometry dedicates a fair amount of its zero-th chapter to examples of (real) differentiable manifolds.  I wanted to add a few more examples, in addition to repeating some of those already mentioned.

Example 1. First, of course, are the Euclidean spaces, \mathbb{R}^n.  Each of these can be covered by the single open set U = \mathbb{R}^n where the parameterization \bold{x}:U\to\mathbb{R}^n is the identity function \bold{x} = \text{id}.  This is a rather trivial example, one that is usually encountered in a multivariable or vector calculus course (without terms like “manifold” or “differentiable structure,” course).

However, it should be noted that there are other differentiable structures one can place on these spaces.  Exercise 11 (page 33) gives two different differentiable structures on the one-dimensional manifold \mathbb{R}, and this can be generalized to the spaces \mathbb{R}^n.  In each case, though, these different manifolds are diffeomorphic; indeed, it can be proven that any two C^{\infty} structures on $\latex \mathbb{R}^n$ are diffeomorphic, except when n = 4. In our seminar, whenever we refer to \mathbb{R}^n as a manifold, it will be with respect to the “trivial” or obvious structure.

Example 2. Any open subset W \subseteq \mathbb{R}^n can be equipped with a differentiable manifold structure by using the (trivial) manifold structure of the ambient space \mathbb{R}^n.  In particular, we can cover W by itself and again use the identity mapping as a parmaterization.  Moreover, given any manifold M and any open subset W \subseteq M, one can equip W with a similarly-defined, inherited manifold structure. This example may seem rather trivial, but it provides us with some pretty important manifolds, most notably, the General Linear group of n\times n real matrices:

\displaystyle \text{GL}_n(\mathbb{R}) = \left\{\text{invertible } n\times n \text{ matrices over } \mathbb{R}\right\}

We can equip \text{GL}_n with a differentiable structure by understanding it as an open subset of \text{GL}(n) \subset \mathbb{R}^{n^2}.  Indeed, it is not too difficult to think of an n\times n matrix as a point in \mathbb{R}^{n^2}; that is, we can identify the set of all n\times n matrices with this Euclidean space, i.e.

\displaystyle \text{M}_n(\mathbb{R}) = \mathbb{R}^{n^2}

The subset portion of this claim is rather clear, then.  That the points of \text{GL}_n can be understood as an open subset perhaps requires a bit more thought.  To argue this point, note that the determinant function \det:\text{M}_n(\mathbb{R})\to\mathbb{R} is continuous and so the set \text{GL}_n = \det^{-1}\left(\mathbb{R}-\{0\}\right) is the inverse-image of an open set under a continuous mapping — it is therefore open.

The set of invertible matrices is therefore a manifold.  Moreover, one notes that these matrices (in addition to being added and rescaled) can be multiplied by one another, and this multiplication process is itself differentiable.  In the n = 2 case, for example, this amounts to noticing that

\displaystyle \left(\bold{x}, \bold{y}\right) \mapsto \left(x_1y_1+x_1y_2, x_3y_1 + x_4y_2, x_1y_3 + x_2y_4, x_3y_3 + x_4y_4\right)

is a bijective, differentiable function (with differentiable inverse) where \bold{x}, \bold{y} \in \mathbb{R}^4.  Moreover, one can check that sending an element \bold{x} \in M_n(\mathbb{R}) to its inverse is also an invertible, differentiable function.  These are the defining properties of what mathematicians and physicists call a Lie Group.

Example 3. Given a manifold M one can construct submanifolds using differentiable functions f:M\to\mathbb{R}.  Given a constant value c \in \mathbb{R} the level set

f^{-1}(c) = \{ p \in M : f(p) = c\} \subseteq M

can often be given the structure of a submanifold.  If the tangent mapping df_p : T_p M\to T_{f(p )}\mathbb{R} is surjective for each p \in f^{-1}(c), for instance, then the level set is a submanifold.  For instance, if we set M = \mathbb{R}^n and use f(\bold{x}) = \|\bold{x}\|^2, then the level set

f^{-1}(1) = \{\bold{x} \in \mathbb{R}^n : \|\bold{x}\|^2 = 1\} = S^n

is the unit sphere.  (As explained by Do Carmo, explicit coordinate charts can be provided for this set.  In fact, it is possible to cover a sphere with two coordinate charts whose images overlap in a connected region, and so these manifolds are, as a result of Example 4.5 (pg. 19) also orientable.)

Example 4. We can combine the previous examples — that of \text{GL}_n(\mathbb{R}) and level sets as submanifolds to produce some other examples.  In particular, we can consider the function f:M_n(\mathbb{R})\to\mathbb{R} given by f(\bold{x}) = \det\bold{x} where, again, we are treating \bold{x} \in \mathbb{R}^{n^2} = M_n(\mathbb{R}).  If we can argue that for every \bold{x} \in f^{-1}(1) the mapping df_p is surjective, then it will immediately follow that this level set is a submanifold of \text{GL}_n(\mathbb{R}).  Some creative computing confirms that, indeed, this is the case since, for instance, one can argue

\displaystyle \frac{\partial \det A}{\partial A_{i\,j}} = \text{adj}^T(A)_{i\,j}

where \text{adj} denotes the adjugate matrix of A.  While there are more direct methods for verifying that the set of determinant-1 matrices forms a sub manifold of invertible matrices, when n = 2 this can all be calculated and verified rather easily.  If we set \bold{p} = (a, b, c, d) \in \text{GL}_2(\mathbb{R}) \subset \mathbb{R}^4, for instance, then we have that df_p(\bold{v}) is given by

\displaystyle df_{p}(\bold{v}) = (d, -c, -b, a)\cdot(v_1, v_2, v_3, v_4)

The only way the map df_p:T_pM \to T_{f(p )}\mathbb{R} \cong \mathbb{R} can fail to be surjective is if df_p is the zero map.  However, this would imply that d = c = b = a = 0 which would violate the assumption that \bold{p} \in \text{GL}_n(\mathbb{R}) represents an invertible matrix.  Therefore, the set of all 2\times2 matrices with determinant equal to 1 forms a submanifold of the general linear group, and it is called the special linear group.  In arbitrary dimension n, it is denoted by \text{SL}_n(\mathbb{R}):

\displaystyle \text{SL}_n(\mathbb{R}) = \left\{\bold{x} \in \text{GL}_n(\mathbb{R}) : \det\bold{x} = 1\right\}.

Because \text{GL}_n(\mathbb{R}) is an open subset of \mathbb{R}^n, we have at once that \dim\text{GL}_n(\mathbb{R}) = n, but what is the dimension of \text{SL}_n(\mathbb{R})?  Because we chose the “level set route” for defining this manifold, the answer is free: if M is a manifold and N \subset M is a submanifold obtained as a level set of a function f:M\to\mathbb{R}, then \dim N = \dim M - 1.  We say that such manifolds are codimension one submanifolds, and so it must follow that \dim \text{SL}_n(\mathbb{R}) = \dim \text{GL}_n(\mathbb{R})-1 = n^2-1.  We will revisit this dimension count later in our seminar, confirming it by determining what the tangent spaces of the special linear group are.

Example 5. Pages 4-5 of Do Carmo explore an important collection of examples: real projective spaces.  These are defined as a set of equivalent classes of points in \mathbb{R}^{n+1} where

\displaystyle [\bold{x}] = \{ \lambda\bold{x} : \lambda \neq 0, \bold{x} \in \mathbb{R}^{n+1} \}

One can also think of this as the set of lines in \mathbb{R}^{n+1} that pass through the origin; equivalently, one can think of this as the set of 1-dimensional subspaces of \mathbb{R}^{n+1}.  However one thinks of these projective spaces, they are all equipped with a differentiable structure in the same way.

Following the explanation offered by Do Carmo, one first breaks this large collection into some easier-to-think-about subsets, ones that are easily mapped to \mathbb{R}^n.  An element [\bold{x}] cannot be represented by the zero vector, and so one of the entries in any of its representatives must be non-zero.  For simplicity, let’s say that x_1 \neq 0, and so we conclude that

\displaystyle [\bold{x}] = [x_1, x_2, \dots, x_{n+1}] = \left[ 1, \frac{x_2}{x_1}, \dots, \frac{x_{n+1}}{x_1}\right]

where we have used \lambda = 1/x_1.  Note that we don’t “need” all n+1 bits of information to understand this point, we only need the n-tuple (x_2/x_1, x_3/x_1, \dots, x_{n+1}/x_1) \in \mathbb{R}^n.  In fact, if we gather together all of the points [\bold{x}] with x_1 \neq 0 into one single subset — called, say V_1 — then such a subset feels an awful lot like \mathbb{R}^n.  This is true for corresponding subsets V_2, V_3, \cdots, V_{n+1}.  In general, one builds functions \bold{x}_i : \mathbb{R}^n \to V_i given by

\displaystyle \bold{x}_i(y_1, y_2, \dots, y_i, \dots y_n) = [y_1, y_2, \dots, y_{i-1}, 1, y_i, y_{i+1}, \dots, y_n].

One readily checks that each function \bold{x}_i is bijective, and a straightforward computation confirms that the transition functions \bold{x}_i^{-1} \circ \bold{x}_j are differentiable functions.

Note that one can also obtain these projective spaces by instead working with an equivalence relation on the n-dimensional sphere S^n \subset \mathbb{R}^{n+1}:

\displaystyle \bold{x}\sim\bold{y} \iff \bold{x} = \pm\bold{y}.

 As Do Carmo remarks, this construction can be generalized using an abstract manifold M and an abstract group G.  M takes the place of the manifold \mathbb{R}^{n+1}, and the group G takes the place of the set \{\lambda \in \mathbb{R} : \lambda \neq 0\} = \mathbb{R}^*, which is a group under multiplication.  In the general setting, the elements of G will act as diffeomorphisms on the points of M, with group multiplication corresponding to the composition of diffeomorphisms.  I will talk more about this in a blog post on Quotient Manifolds.  For now, though, note that this abstract machinery will allow us to easily place a differentiable structure on so-called n-dimensional tori that are defined as equivalence classes

\displaystyle T^n = \mathbb{R}^n / \mathbb{Z}^n = \{[\bold{x}] : \bold{x} \in \mathbb{R}^n\}

where \bold{y} \in [\bold{x}] \iff \bold{x}-\bold{y} \in \mathbb{Z}^n.  The group G = \mathbb{Z}^n acts on the manifold M = \mathbb{R}^n via addition — that is a point g = (z_1, \dots z_n) = \bold{z} \in G acts on \bold{x} as follows:

\displaystyle g(\bold{x}) = \bold{x} + \bold{z}

Torus_from_rectangleand so [\bold{x}] contains all integer translates of the point \bold{x}.

When n = 2, this set of equivalence classes produces the familiar (topological) picture of the surface of a doughnut, obtained by gluing the correctly-identified edges of a square.  This animation to the left displays this process quite nicely.

Example 6.  One can generalize real projective spaces in other important ways, too.   This includes starting with the manifold M = \mathbb{R}^n, but instead letting \text{G}(k, n) denote the set of all k-dimensional subspaces of M (where k \leq n).  There is a standard way to equip \text{G}(k, n) with a differentiable structure (in fact, it becomes another Lie Group), and once this is done the resulting manifold is known as a Grassmannian.

Notes on Topology

Insights and theorems from point-set topology will be used throughout this seminar, especially those concerning manifolds.  However, we will minimize their prominence in this seminar.  Keep in mind that topology is always present, though, lurking in the shadows.

Do Carmo manages to cover quite a bit of material before admitting this, too.  On page 29 he notes that “Up till now we have put no restrictions on the topology of a differentiable manifold.  In fact, the natural topology of a manifold can be quite strange.”  Indeed, little to no (overt) use of topology has been made up until this point.  Save for Remark 2.3 (on page 3), little attention is paid to a smooth manifold’s topological properties.  As mentioned back on page 29, though, these properties can be a bit weird or lacking.

The two basic topological properties that can fail for manifolds are (A) the Hausdorff property and (B) the Countable Basis property.  Do Carmo notes that (A) is needed for the existence of unique limits (important given that we want to take limits, i.e. differentiate) and that (B) is needed to build something called a differentiable partition of unity.

Very soon we will equip our tangent spaces with inner products (called a Riemannian metric), and the existence of such an object is usually proven via a partition of unity argument.  These are important but somewhat technical points that we spend class time discussing, however I will bring them up occasionally in blog posts and remain happy to discuss them during office hours.

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