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

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Typed Notes - First Hour

Reminder:

1) An immersion locally looks like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n\rightarrow\mathbb{R}^m} given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\mapsto(x,0)}

2) A submersion locally looks like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^m\rightarrow\mathbb{R}^n} given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y)\mapsto x}

Today's Goals

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

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

Definition 1

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:M^m\rightarrow N^n} be smooth. A point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in M^m} is critical if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_p} is not onto Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow} rank $df_{p}<n$. Otherwise, p is regular.

Definition 2 A point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\in N^n} is a critical value of f if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists p\in M^m} such that p is critical and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(p) = y} . Otherwise, y is a regular value

Example 1

Consider the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:S^2\rightarrow\mathbb{R}^2} given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,z)\mapsto x} .

SoFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_1 + tv_2\}} but implicitly they are the solutions to the equations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax+by+cz = d} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} ) is a "good" intersection as it IS stable under small perturbations.

Precisely,

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1^{n_1}, N_2^{n_2} \subset M} be smooth submanifolds. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in N_1^{n_1} \bigcap N_2^{n_2}}

We say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1} is transverse to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_2} in M at p if for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N_1\subset T_p M} andFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1} intersects Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_2} transversally if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1} is transversal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_2} at every point

Theorem 3

If $N_{1}^{n_{1}}\bigcap N_{2}^{n_{2}}\subset M$ transversally then

1) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1\bigcap N_2} is a manifold of dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_1 + n_2 - m}

2) Locally can find charts so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1 = \{(n_1\ arb.\ pts.),( m-n_1\ zeros)\}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_2 = \{(m-n_2\ zeros),( m-n_2\ arb.\ pts.)\}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2} subspaces of V then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dim(W_1 + W_2) + dim(W_1\bigcap W_2) = dim(W_1) + dim)W_2)}

In particular if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1 + W_2 = V} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dim(W_1\bigcap W_2) = -dim(V) + dim(W_1) + dim(W_2)}

Proof Scheme for Theorem 3

We can write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_i = \varphi_i^{-1}(0)} for some such Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_i:U\subset M \rightarrow\mathbb{R}^{m-n_1 = cod(N_i)}}

We then write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1\bigcap N_2 = \varphi^{-1}(0)}

We want Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ker(d\varphi) = ker(d\varphi_1)\bigcap ker(d\varphi_2)}

hence, and by rank nullity, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle rank(d\varphi_p) = m - dim(ker(d\varphi)) = m - \left( dim(TN_1) + dim(TN_2) - dim(TN_1 + TN_2)\right) = }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1\bigcap N_2} is a submanifold.

Now, we know we can construct the following diagram,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} N_1\bigcap N_2 & \rightarrow^{\varphi} & \mathbb{R}\\ \downarrow &&\downarrow^{\iota}\\ M&\rightarrow^{\lambda} & \mathbb{R}^m\\ \end{matrix}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iota(x) = (x,0)}

We then set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi =} (the function that takes the first Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_1 +n_2-m} coordinates only)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \circ\lambda}

hence, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi: M\rightarrow\mathbb{R}^{n_1 + n_2 - m}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi |_{N_1\bigcap N_2}} is a chart for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1\bigcap N_2}

We now consider Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta: M\rightarrow\mathbb{R}^m} given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\varphi_2,\psi, \varphi_1)}

i.e. operating via the following table,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} &N_1&N_2\\ \varphi_2&\zeta&0\\ \psi&\zeta&\zeta\\ \varphi_1&0&\zeta\\ \end{matrix}}

Then,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 4

Consider a standard "first year" smooth function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 5

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) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists f:\mathbb{R}\rightarrow\mathbb{R}} whose critical values are homeomorphic to a canter set.

2) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists} cantor sets with measure arbitrarily close to 1

3) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^{\infty}} in the theorem.

Evil functions

Example 1

There exists a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:\mathbb{R}\rightarrow\mathbb{R}} smooth such that its set of critical values is homeomorphic to a Cantor set.

Remember that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_{a,b}:[a,b]\rightarrow \amthbb{R}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_{a,b}(x)=e^{-1/(x-a)^2}e^{-1/(x-b)^2}}

is a smooth function such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_{a,b}(a)=g_{a,b}(b)=0}

We can define the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} in the complement of a Cantor set using the appropriate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_{a,b}} in its of the intervals.

Notices that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\int_{0}^{x}h(t)dt} holds the conditions of the example.