0708-1300/Class notes for Tuesday, October 9

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Class Notes

The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.

Scanned Notes

Scanned notes for today's class can be found here

(see discussion)


Typed Notes - First Hour

Reminder:

1) An immersion locally looks like \mathbb{R}^n\rightarrow\mathbb{R}^m given by x\mapsto(x,0)

2) A submersion locally looks like \mathbb{R}^m\rightarrow\mathbb{R}^n given by (x,y)\mapsto x


Today's Goals

1) More about "locally things look like their differential"

2) The tricky Sard's Theorem: "Evil points are rare, good points everywhere"


Definition 1

Let f:M^m\rightarrow N^n be smooth. A point p\in M^m is critical if df_p is not onto \Leftrightarrow rank df_p <n. Otherwise, p is regular.


Definition 2 A point y\in N^n is a critical value of f if \exists p\in M^m such that p is critical and f(p) = y. Otherwise, y is a regular value


Example 1

Consider the map f:S^2\rightarrow\mathbb{R}^2 given by (x,y,z)\mapsto (x,y). I.e., the projection map. The regular points are all the points on S^2 except the equator. The regular values, however, are all (x,y) such that x^2+y^2 \neq 1


Example 2

Consider f:\mathbb{R}^3\rightarrow\mathbb{R} given by p\mapsto ||p||^2. That is, (x,y,z)\mapsto x^2+y^2+z^2. Clearly df|_{p=(x,y,z)} = (2x,2y,2z) and so p is regular \Leftrightarrow df_p\neq 0 \Leftrightarrow p\neq 0

So, the critical values are the image of zero, thus only zero. All other x\in\mathbb{R} are regular values.


Note: In both the last two examples there were points in the target space that were NOT hit by the function and thus are vacuously regular. In the previous example these are the point x<0.


Example 3

Consider a function \gamma from a segment in \mathbb{R} onto a curve in \mathbb{R}^2 such that d\gamma is never zero. Thus, rank(d\gamma = 1) and so d\gamma is never onto. Hence, ALL points are critical in the segment. The points on the curve are critical values, as they are images of critical points, and all points in \mathbb{R}^2 NOT on the curve are vacuously regular.


Theorem 1

Sard's Theorem

Almost every y\in N^n is regular \Leftrightarrow the set of critical values of f is of measure zero.


Note: The measure is not specified (indeed, for a topological space there is no canonical measure defined). However the statement will be true for any measure.


Theorem 2

If f:M^m\rightarrow N^n is smooth and y is a regular value then f^{-1}(y) is an embedded submanifold of M^m of dimension m-n.


Re: Example 2

f^{-1}(y) is a sphere and hence (again!) the sphere is a manifold


Re: Example 3

f^{-1}(y) for regular y is empty and hence we get the trivial result that the empty set is a manifold


Proof of Theorem 2

Let f:M^m\rightarrow N^n is smooth and y is a regular value. Pick a p\in f^{-1}(y). p is a regular point and thus df_p is onto. Hence, by the submersion property (Reminder 2) we can find a "good charts" thats maps a neighborhood U of p by projection to a neighborhood V about y. Indeed, on U f looks like \mathbb{R}^n\times\mathbb{R}^{m-1}\rightarrow \mathbb{R}^n by (x,z)\mapsto x.

So f^{-1}(0) = \{(0,z)\} = \mathbb{R}^{m-n}. Q.E.D


Diversion

Arbitrary objects can be described in two ways:

1) With a constructive definition

2) with an implicit definition

For example, a constructive definition of lines in \mathbb{R}^3 is given by \{v_1 + tv_2\} but implicitly they are the solutions to the equations ax+by+cz = d and ez+fy+gz+h.

Hence in general, a constructive definition can be given in terms of an image and an implicit definition can be given in terms of a kernal.


Homological algebra is concerned with the difference between these philosophical approaches.


Remark

For submanifolds of smooth manifolds, there is no difference between the methods of definition.


Definition 3

Loosely we have the idea that a concave and convex curve which just touch at a tangent point is a "bad" intersection as it is unstable under small perturbation where as the intersection point in an X (thought of as being in \mathbb{R}^2) is a "good" intersection as it IS stable under small perturbations.

Precisely,

Let N_1^{n_1}, N_2^{n_2} \subset M be smooth submanifolds. Let p\in N_1^{n_1} \bigcap N_2^{n_2}

We say N_1 is transverse to N_2 in M at p if for T_p N_1\subset T_p M and T_p N_2 \subset T_p M satisfies T_p N_1 + T_p N_2 = T_p M


Example 4

Our concave intersecting with convex curve example intersecting tangentially has both of their tangent spaces at the intersection point being the same line and thus does not intersect transversally as the sum of the tangent spaces is not all of \mathbb{R}^2.

Our X example does however work.

Typed Notes - Second Hour

Assistance needed: There is a symbol for "intersects transversally" but I am not sure of the latex command. See the scanned notes for what this looks like, I will just write it.


Definition 4

N_1 intersects N_2 transversally if N_1 is transversal to N_2 at every point


Theorem 3

If N_1^{n_1}\bigcap N_2^{n_2} \subset M transversally then

1) N_1\bigcap N_2 is a manifold of dimension n_1 + n_2 - m

2) Locally can find charts so that N_1 = \{(n_1\ arb.\ pts.),( m-n_1\ zeros)\} and N_2 = \{(m-n_2\ zeros),( m-n_2\ arb.\ pts.)\} with N_1\bigcap N_2 = \{(m-n_2\ arb.\ pts.),(n_1+n_2 - 2m\ zeros), (m-n_1\ arb.\ pts.)\}


Recall a Thm from Linear Algebra

If W_1 and W_2 subspaces of V then dim(W_1 + W_2) + dim(W_1\bigcap W_2) = dim(W_1) + dim)W_2)

In particular if W_1 + W_2 = V then dim(W_1\bigcap W_2) = -dim(V) + dim(W_1) + dim(W_2)


Proof Scheme for Theorem 3

We can write N_i = \varphi_i^{-1}(0) for some such \varphi_i:U\subset M \rightarrow\mathbb{R}^{m-n_1 = cod(N_i)}

We then write \varphi = \varphi_1\times\varphi_2: M\rightarrow \mathbb{R}^{m-n_1}\times\mathbb{R}^{m-n_2} = \mathbb{R}^{2m - n_1 - n_2}

Hence, N_1\bigcap N_2 = \varphi^{-1}(0)

We want rank(d\varphi_p) = 2m-n_1-n_2. To prove this, we consider the aforementioned theorem from linear algebra with respect to the vector spaces obeying ker(d\varphi) = ker(d\varphi_1)\bigcap ker(d\varphi_2)

hence, and by rank nullity, rank(d\varphi_p) = m - dim(ker(d\varphi)) = m - \left( dim(TN_1) + dim(TN_2) - dim(TN_1 + TN_2)\right) =

m - (n_1 + n_2 - m) = 2m- n_1 - n_2 as we wanted.

This shows that 0 is a regular point and hence by our previous theorem N_1\bigcap N_2 is a submanifold.

Now, we know we can construct the following diagram,

\begin{matrix}
N_1\bigcap N_2 & \rightarrow^{\varphi} & \mathbb{R}\\
\downarrow &&\downarrow^{\iota}\\
M&\rightarrow^{\lambda} & \mathbb{R}^m\\
\end{matrix}

where \iota(x) = (x,0)

We then set \psi = (the function that takes the first n_1 +n_2-m coordinates only)\circ\lambda

hence, \psi: M\rightarrow\mathbb{R}^{n_1 + n_2 - m} and \psi |_{N_1\bigcap N_2} is a chart for N_1\bigcap N_2

We now consider \zeta: M\rightarrow\mathbb{R}^m given by (\varphi_2,\psi, \varphi_1)

i.e. operating via the following table,

\begin{matrix}
&N_1&N_2\\
\varphi_2&\zeta&0\\
\psi&\zeta&\zeta\\
\varphi_1&0&\zeta\\
\end{matrix}

Then,

d\zeta = \begin{bmatrix}
I&0\\
I&I\\
0&I\\
\end{bmatrix}

for blocks I of the appropriate sizes.

Thus (loosely) Q.E.D


Now on to some examples and comments about why Sard's Theorem is expected, but not obvious:

Example 5

Consider a standard "first year" smooth function f:\mathbb{R}\rightarrow\mathbb{R}. The "critical points" are in the first year calculus sense where the derivative is zero and the critical values are the images of these points. hence, the set of critical values we expect to be "small"

Example 6

Consider the function that folds the plane in half. The critical points are along the fold, as are the critical values and this line has 1 dimension and so of trivial measure in the plane (not that we have given it a measure yet!)


Claims:

1) \exists f:\mathbb{R}\rightarrow\mathbb{R} whose critical values are homeomorphic to a canter set.

2) \exists cantor sets with measure arbitrarily close to 1

3) \exists g\in C^1:\mathbb{R}^2\rightarrow\mathbb{R} whose critical points are a cantor set cross a cantor set and whose critical values are everything. Hence we will need out functions to be C^{\infty} in the theorem.

Evil functions

Example 1

There exists a function f:\mathbb{R}\rightarrow\mathbb{R} smooth such that its set of critical values is homeomorphic to a Cantor set.

Remember that g_{a,b}:[a,b]\rightarrow \mathbb{R}

g_{a,b}(x)=e^{-1/(x-a)^2}e^{-1/(x-b)^2}

is a smooth function such that g_{a,b}(a)=g_{a,b}(b)=0

We can define the function h in the complement of a Cantor set using the appropriate g_{a,b} in its of the intervals.

Notices that f(x)=\int_{0}^{x}h(t)dt holds the conditions of the example.