0708-1300/Class notes for Tuesday, September 25

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Dror's Notes

• Class photo is on Thursday, show up and be at your best! More seriously -
  • The class photo is of course not mandatory, and if you are afraid of google learning about you, you should not be in it.
  • If you want to be in the photo but can't make it on Thursday, I'll take a picture of you some other time and add it as an inset to the main picture.

• I just got the following email message, which some of you may find interesting:

NSERC - CMS Math in Moscow Scholarships

The Natural Sciences and Engineering Research Council (NSERC) and the
Canadian Mathematical Society (CMS) support scholarships at $9,000
each. Canadian students registered in a mathematics or computer
science program are eligible.

The scholarships are to attend a semester at the small elite Moscow
Independent University.

Math in Moscow program
www.mccme.ru/mathinmoscow/
Application details
www.cms.math.ca/bulletins/Moscow_web/

For additional information please see your department or call the CMS
at 613-562-5702.

Deadline September 30, 2007 to attend the Winter 2008 semester.

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.

First hour

Recall from last class we were proving the equivalence of the two definitions for a smooth manifold. The only nontrivial point that remained to be proved was that if we started with the definition of a manifold in the sense of functional structures and produced charts 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,\psi} that these charts would satisfy the property of a manifold, defined in the atlas sense, 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 \psi\circ\varphi^{-1}} is smooth where defined.

Proof

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\circ\varphi^{-1}} :Rⁿ$\rightarrow$Rⁿ is smooth 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} 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\circ\varphi^{-1})_i} :RⁿFailed 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 \rightarrow} R is smooth 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 \forall} i 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} 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 \pi_i\circ\psi\circ\varphi^{-1}} is smooth 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 \pi_i} is the $i^{th}$ coordinate projection map.

Now, since $\pi _{i}$ is always smooth, $\pi _{i}\in F_{R^{n}}(U_{\psi }^{'})$

But then we have $\pi _{i}\circ \psi \in F_{M}(U_{\psi })$ and so, by a property of functional structures, $\pi _{i}\circ \psi |_{U_{\varphi }\bigcap U_{\psi }}\in F_{M}(U_{\varphi }\bigcap U_{\psi })$ and hence $\pi _{i}\circ \psi \circ \varphi ^{-1}\in F_{R^{n}}$ where it is defined and thus is smooth. QED

Definition 1 (induced structure) Suppose $\pi :X\rightarrow Y$ and suppose Y is equipped with a functional structure $F_{Y}$ then the "induced functional structure" on X is

$F_{X}(U)=\{f:U\rightarrow R\ |\ \exists g\in F_{Y}(V)\ such\ that\ V\supset \pi (U)\ and\ f=g\circ \pi \}$

Claim: this does in fact define a functional structure on X

Definition 2 This is the reverse definition of that given directly above. Let $\pi :X\rightarrow Y$ and let X be equipped with a functional structure $F_{X}$. Then we get a functional structure on Y by $F_{Y}(V)=\{g:V\rightarrow R\ |\ g\circ \pi \in F_{X}(\pi ^{-1}(V)\}$ Claim: this does in fact define a functional structure on X

Example 1 Let $S^{2}=R^{3}-\{0\}/$~ where the equivalence relation ~ is given by x~$\alpha$x for $\alpha$>0 We thus get a canonical projection map $\pi :R^{3}-\{0\}\rightarrow S^{2}$ and hence, there is an induced functional structure on $S^{2}$. Claim: 1) This induced functional structure makes 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 S^{2}} into a manifold 2) This resulting manifold is the same manifold as from the atlas definition given previously

Example 2 Consider the torus thought of as 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^{2} = R^{2}/Z^{2}} , i.e., the real plane with the equivalence relation that (x,y)~(x+n,y+m) for (x,y) 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 R^{2}} and (n,m) 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 Z^{2}}

As in the previous example, the torus inherits a functional structure from the real plane we must again check that 1) We get a manifold 2) This is the same manifold as we had previously with the atlas definition

Example 3 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 CP^{n}} denote the n dimensional complex projective space, that is, 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 CP^{n} = C^{n+1}-\{0\}/} ~ 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 [z_{0},...,z_{n}]} ~ 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 [\alpha z_{0},...,\alpha z_{n}]} 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 \alpha\in C}

Again, this space inherits a functional structure from $C^{n+1}$ and we again need to claim that this yields a manifold.

Proof of Claim

We consider the subsets 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 CP^{n}\supset U_{i} = \{[z_{0},...,z_{n}]\ |\ z_{i}\neq 0\}} 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 0\leq i \leq n}

Clearly 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 \bigcup U_{i} = CP^{n}}

Now, for each 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 U_{i}} there is a unique representative for its equivalence class of the form 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 [z_{0},...,1,...,z_{n}]} where the 1 is at the ith location.

We thus can get a map from 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_{i}\rightarrow C^{n} = R^{2n}} 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 p\mapsto [z_{0}/z_{i},...,z_{i-1}/z_{i},z_{i+1}/z_{i},...,z_{n}/z_{i}]} Hence we have shown (loosely) that our functional structure is locally isormophic 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 (R,C^\infty)}

Definition 3 Product Manifolds

Suppose 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^{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 N^{n}} are manifolds. Then the product manifold, on the set MxN has an atlas 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 \times \psi: U\times V\rightarrow U'\times V'\in R^{m}\times R^{n}\ | \varphi: U\rightarrow U'\subset R^{m}\ and\ \psi:V\rightarrow V'\subset R^{n}} are charts in resp. manifolds}

Claim: This does in fact yield a manifold

Example 4 It can be checked 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 T^{2} = S^{1}\times S^{1}} gives the torus a manifold structure, by the product manifold, that is indeed the same as the normal structure given previously.