0708-1300/Class notes for Tuesday, October 2 - Revision history

Jvoltz at 21:14, 6 November 2007

2007-11-06T21:14:05Z

← Older revision		Revision as of 17:14, 6 November 2007
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	Locally, every immersion looks like the inclusion <math>\iota:\mathbb{R}^m\rightarrow \mathbb{R}^n</math>.		Locally, every immersion looks like the inclusion <math>\iota:\mathbb{R}^m\rightarrow \mathbb{R}^n</math>.

	More precisely, if <math>\theta:M^m\rightarrow N^n</math> and <math>d\theta_p</math> is 1:1 then there exist charts <math>\varphi</math> acting on <math>U\subset M</math> and <math>\psi</math> acting on <math>V\subset N</math> such that for <math>p\in U, \~~phi~~(p) = \psi(\theta(p)) = 0</math> such that the following diagram commutes:		More precisely, if <math>\theta:M^m\rightarrow N^n</math> and <math>d\theta_p</math> is 1:1 then there exist charts <math>\varphi</math> acting on <math>U\subset M</math> and <math>\psi</math> acting on <math>V\subset N</math> such that for <math>p\in U, \varphi(p) = \psi(\theta(p)) = 0</math> such that the following diagram commutes:


	<math>\begin{matrix}		<math>\begin{matrix}
	U&\rightarrow^{\~~phi~~}&U'\subset \mathbb{R}^m\\		U&\rightarrow^{\varphi}&U'\subset \mathbb{R}^m\\
	\downarrow_{\theta} &&\downarrow_{\iota} \\		\downarrow_{\theta} &&\downarrow_{\iota} \\
	V& \rightarrow^{\psi}& V'\subset \mathbb{R}^n\\		V& \rightarrow^{\psi}& V'\subset \mathbb{R}^n\\

Megan: /* Class Notes */

2007-11-03T20:37:25Z

Class Notes

← Older revision		Revision as of 16:37, 3 November 2007
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	Locally, every immersion looks like the inclusion <math>\iota:\mathbb{R}^m\rightarrow \mathbb{R}^n</math>.		Locally, every immersion looks like the inclusion <math>\iota:\mathbb{R}^m\rightarrow \mathbb{R}^n</math>.

	More precisely, if <math>\theta:M^m\rightarrow N^n</math> and <math>d\theta_p</math> is 1:1 then there ~~exists~~ charts <math>\varphi</math> acting on <math>U\subset M</math> and <math>\~~phi~~</math> acting on <math>V\subset N</math> such that for <math>p\in U, \phi(p) = \psi(\theta(p)) = 0</math> such that the following diagram commutes:		More precisely, if <math>\theta:M^m\rightarrow N^n</math> and <math>d\theta_p</math> is 1:1 then there exist charts <math>\varphi</math> acting on <math>U\subset M</math> and <math>\psi</math> acting on <math>V\subset N</math> such that for <math>p\in U, \phi(p) = \psi(\theta(p)) = 0</math> such that the following diagram commutes:

Trefor: /* Second Hour */

2007-11-03T02:24:40Z

Second Hour

← Older revision		Revision as of 22:24, 2 November 2007
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	''Proof of Theorem 2''		''Proof of Theorem 2''

	We start with the normal situation of <math>\theta:M\rightarrow N</math> with M,N manifolds with atlases containing <math>(\varphi_0,~~U_)0~~)</math> and <math>(\psi_0, V_0)</math> respectively. We also expect that for <math>p\in U_0, \varphi_0(p) = \psi_0(\theta(p)) = 0</math>. I will first draw the diagram and will subsequently justify the relevant parts. The proof reduces to showing a certain part of the diagram commutes appropriately.		We start with the normal situation of <math>\theta:M\rightarrow N</math> with M,N manifolds with atlases containing <math>(\varphi_0,U_0)</math> and <math>(\psi_0, V_0)</math> respectively. We also expect that for <math>p\in U_0, \varphi_0(p) = \psi_0(\theta(p)) = 0</math>. I will first draw the diagram and will subsequently justify the relevant parts. The proof reduces to showing a certain part of the diagram commutes appropriately.

Trefor: /* First Hour */

2007-11-02T23:13:49Z

First Hour

← Older revision		Revision as of 19:13, 2 November 2007
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	\end{bmatrix}</math>		\end{bmatrix}</math>

	where the 1 is at the kth location. In other words, <math>\theta_*D = \sum_{j=1}^{n} \sum_{i=1}^{m}~~a_j~~ \frac{\partial \~~theta_i~~}{\partial ~~x_j~~} \frac{\partial }{\partial ~~y_i~~} </math>		where the 1 is at the kth location. In other words, <math>\theta_*D = \sum_{j=1}^{n} \sum_{i=1}^{m}a_i \frac{\partial \theta_j}{\partial x_i} \frac{\partial }{\partial y_j} </math>

Trefor: /* First Hour */

2007-11-02T22:40:47Z

First Hour

← Older revision		Revision as of 18:40, 2 November 2007
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	7) We can consider operators on functions to be in a sense dual to the functions and hence should go in the opposite direction. Hence, tangent vectors, defined in the sense of derivations, ''pushforward'' via <math>D\rightarrow (\theta_D)(f):= D(\theta^f) = D(f\circ\theta</math>		7) We can consider operators on functions to be in a sense dual to the functions and hence should go in the opposite direction. Hence, tangent vectors, defined in the sense of derivations, ''pushforward'' via <math>D\rightarrow (\theta_D)(f):= D(\theta^f) = D(f\circ\theta)</math>

	CHECK: This definition satisfies linearity and Liebnitz property.		CHECK: This definition satisfies linearity and Liebnitz property.

Trefor: /* First Hour */

2007-11-02T22:40:26Z

First Hour

← Older revision		Revision as of 18:40, 2 November 2007
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	7) We can consider operators on functions to be in a sense dual to the functions and hence should go in the opposite direction. Hence, tangent vectors, defined in the sense of derivations, ''pushforward'' via <math>D\rightarrow (\theta_D)(f):= D(\theta^f)</math>		7) We can consider operators on functions to be in a sense dual to the functions and hence should go in the opposite direction. Hence, tangent vectors, defined in the sense of derivations, ''pushforward'' via <math>D\rightarrow (\theta_D)(f):= D(\theta^f) = D(f\circ\theta</math>

	CHECK: This definition satisfies linearity and Liebnitz property.		CHECK: This definition satisfies linearity and Liebnitz property.

Bpym: /* First Hour */

2007-10-09T01:26:59Z

First Hour

← Older revision		Revision as of 21:26, 8 October 2007
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	Now, for a <math>D\in T_pM</math> we can write <math>D=\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}</math>		Now, for a <math>D\in T_pM</math> we can write <math>D=\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}</math>

			So,
⚫	~~So,~~ <math>~~(\theta_*D)(f)~~ = ~~\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\varphi) ///\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\theta)should be?? ///=~~ \sum_{i=1}^m a_i \sum_{j=1}^n\frac{\partial f}{\partial y_j}\frac{\partial\theta_j}{\partial x_i}=</math>
			{\|
⚫	<math>=\begin{bmatrix}
			\|-
⚫	\frac{\partial f}{\partial y_1} & \cdots & \frac{\partial f}{\partial y_n}\\
			\|<math>(\theta_*D)(f) \!</math>
⚫	\end{bmatrix}
			\|<math> = D(\theta^* f)\!</math>
⚫	\begin{bmatrix}
			\|-

			\|
⚫	\frac{\partial \theta_1}{\partial x_1} & \cdots & \frac{\partial \theta_1}{\partial x_m}\\
			\|<math>=\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\theta) </math>
⚫	\vdots& & \vdots\\
			\|-
⚫	\frac{\partial \theta_n}{\partial x_1} & \cdots & \frac{\partial \theta_n}{\partial x_m}\\
			\|
⚫	\end{bmatrix}
		⚫	\|<math>=\sum_{i=1}^m a_i \sum_{j=1}^n\frac{\partial f}{\partial y_j}\frac{\partial\theta_j}{\partial x_i} </math>
⚫	\begin{bmatrix}
			\|-
⚫	a_1\\
			\|
⚫	\vdots\\
		⚫	\|<math>=\begin{bmatrix}
	a_m\\
		⚫	\frac{\partial f}{\partial y_1} & \cdots & \frac{\partial f}{\partial y_n}\\
⚫	\end{bmatrix}~~</math>~~
		⚫	\end{bmatrix}
		⚫	\begin{bmatrix}
		⚫	\frac{\partial \theta_1}{\partial x_1} & \cdots & \frac{\partial \theta_1}{\partial x_m}\\
		⚫	\vdots& & \vdots\\
		⚫	\frac{\partial \theta_n}{\partial x_1} & \cdots & \frac{\partial \theta_n}{\partial x_m}\\
		⚫	\end{bmatrix}
		⚫	\begin{bmatrix}
		⚫	a_1\\
		⚫	\vdots\\
			a_m\\
		⚫	\end{bmatrix}
			</math>
			\|}


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	\end{bmatrix}</math>		\end{bmatrix}</math>

			where the 1 is at the kth location. In other words, <math>\theta_*D = \sum_{j=1}^{n} \sum_{i=1}^{m}a_j \frac{\partial \theta_i}{\partial x_j} \frac{\partial }{\partial y_i} </math>
	where the 1 is at the kth location.

Kuramay: /* First Hour */

2007-10-08T07:06:09Z

First Hour

← Older revision		Revision as of 03:06, 8 October 2007
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	Now, for a <math>D\in T_pM</math> we can write <math>D=\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}</math>		Now, for a <math>D\in T_pM</math> we can write <math>D=\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}</math>

	So, <math>(\theta_*D)(f) = \sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\varphi) = \sum_{i=1}^m a_i \sum_{j=1}^n\frac{\partial f}{\partial y_j}\frac{\partial\theta_j}{\partial x_i}=</math>		So, <math>(\theta_*D)(f) = \sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\varphi) ///\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\theta)should be?? ///= \sum_{i=1}^m a_i \sum_{j=1}^n\frac{\partial f}{\partial y_j}\frac{\partial\theta_j}{\partial x_i}=</math>
	<math>=\begin{bmatrix}		<math>=\begin{bmatrix}
	\frac{\partial f}{\partial y_1} & \cdots & \frac{\partial f}{\partial y_n}\\		\frac{\partial f}{\partial y_1} & \cdots & \frac{\partial f}{\partial y_n}\\

Aptikuis: /* Second Hour */

2007-10-06T02:41:48Z

Second Hour

← Older revision		Revision as of 22:41, 5 October 2007
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	'''Example 2'''		'''Example 2'''

	This is the map from R to <math>R^2</math> that looks like a loop-de-loop on a roller coaster (but squashed into the plane of course!) The map <math>\theta</math> itself is NOT 1:1 (consider the crossover point) however <math>\theta_*</math> IS 1:1, hence an immersion.		This is the map from <math>\mathbb{R}</math> to <math>\mathbb{R}^2</math> that looks like a loop-de-loop on a roller coaster (but squashed into the plane of course!) The map <math>\theta</math> itself is NOT 1:1 (consider the crossover point) however <math>\theta_*</math> IS 1:1, hence an immersion.


	'''Example 3'''		'''Example 3'''

	Consider the map from R to <math>R^2</math> that looks like a check mark. While this map itself is 1:1, <math>\theta_*</math> is NOT 1:1 (at the cusp in the check mark) and hence is not an immersion.		Consider the map from <math>\mathbb{R}</math> to <math>\mathbb{R}^2</math> that looks like a check mark. While this map itself is 1:1, <math>\theta_*</math> is NOT 1:1 (at the cusp in the check mark) and hence is not an immersion.


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	'''Example 5'''		'''Example 5'''

	The torus, as a subset of <math>R^3</math> is an immersion		The torus, as a subset of <math>\mathbb{R}^3</math> is an immersion


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	'''Theorem 2'''		'''Theorem 2'''

	Locally, every immersion looks like the inclusion <math>\iota:R^m\rightarrow R^n</math>.		Locally, every immersion looks like the inclusion <math>\iota:\mathbb{R}^m\rightarrow \mathbb{R}^n</math>.

	More precisely, if <math>\theta:M^m\rightarrow N^n</math> and <math>d\theta_p</math> is 1:1 then there exists charts <math>\varphi</math> acting on <math>U\subset M</math> and <math>\phi</math> acting on <math>V\subset N</math> such that for <math>p\in U, \phi(p) = \psi(\theta(p)) = 0</math> such that the following diagram commutes:		More precisely, if <math>\theta:M^m\rightarrow N^n</math> and <math>d\theta_p</math> is 1:1 then there exists charts <math>\varphi</math> acting on <math>U\subset M</math> and <math>\phi</math> acting on <math>V\subset N</math> such that for <math>p\in U, \phi(p) = \psi(\theta(p)) = 0</math> such that the following diagram commutes:
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	<math>\begin{matrix}		<math>\begin{matrix}
	U&\rightarrow^{\phi}&U'\subset R^n\\		U&\rightarrow^{\phi}&U'\subset \mathbb{R}^m\\
	\downarrow_{\theta} &&\downarrow_{\iota} \\		\downarrow_{\theta} &&\downarrow_{\iota} \\
	V& \rightarrow^{\psi}& V'\subset R^n\\		V& \rightarrow^{\psi}& V'\subset \mathbb{R}^n\\
	\end{matrix}</math>		\end{matrix}</math>

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	'''Example 6'''		'''Example 6'''

	The torus is a submanifold as the natural immersion into <math>R^3</math> is 1:1		The torus is a submanifold as the natural immersion into <math>\mathbb{R}^3</math> is 1:1


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	'''Example 7'''		'''Example 7'''

	Consider the map from <math>R\rightarrow R^2</math> whose graph looks like the open interval whose two ends have been wrapped around until they just touch (would intersect at one point if they were closed) the points 1/3 and 2/3rds of the way along the interval respectively.		Consider the map from <math>\mathbb{R}\rightarrow \mathbb{R}^2</math> whose graph looks like the open interval whose two ends have been wrapped around until they just touch (would intersect at one point if they were closed) the points 1/3 and 2/3rds of the way along the interval respectively.
	The map is both 1:1 and an immersion. However, any neighborhood about the endpoints of the interval will ALSO include points near the 1/3rd and 2/3rd spots on the line, i.e., the topology is different and hence this is not an embedding.		The map is both 1:1 and an immersion. However, any neighborhood about the endpoints of the interval will ALSO include points near the 1/3rd and 2/3rd spots on the line, i.e., the topology is different and hence this is not an embedding.

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	The functional structure on an embedded manifold induced by the functional structure on the containing manifold is equal to its original functional structure.		The functional structure on an embedded manifold induced by the functional structure on the containing manifold is equal to its original functional structure.

	Indeed, for all smooth <math>f:M\rightarrow R</math> and <math>\forall p\in M</math> there exists a neighborhood V of <math>\theta(p)</math> and a smooth <math>g:N\rightarrow R</math> such that <math>g\|_{\theta(M)\bigcap U} = f\|_U</math>		Indeed, for all smooth <math>f:M\rightarrow \mathbb{R}</math> and <math>\forall p\in M</math> there exists a neighborhood V of <math>\theta(p)</math> and a smooth <math>g:N\rightarrow \mathbb{R}</math> such that <math>g\|_{\theta(M)\bigcap U} = f\|_U</math>


	''Proof of Corollary 1''		''Proof of Corollary 1''

	Loosely (and a sketch is most useful to see this!) we consider the embedded submanifold M in N and consider its image, under the appropriate charts, to a subset of <math>R^m\subset R^n</math>. We then consider some function defined on M, and hence on the subset in R^n which we can extend canonically as a constant function in the "vertical" directions. Now simply pullback into N to get the extended member of the functional structure on N.		Loosely (and a sketch is most useful to see this!) we consider the embedded submanifold M in N and consider its image, under the appropriate charts, to a subset of <math>\mathbb{R}^m\subset \mathbb{R}^n</math>. We then consider some function defined on M, and hence on the subset in <math>\mathbb{R}^n</math> which we can extend canonically as a constant function in the "vertical" directions. Now simply pullback into N to get the extended member of the functional structure on N.


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	<math>		<math>
	\begin{matrix}		\begin{matrix}
	M\supset U_0 & \rightarrow^{\varphi_0} & U_1\subset R^m & \rightarrow^{Id} & U_2 = U_1 \\		M\supset U_0 & \rightarrow^{\varphi_0} & U_1\subset \mathbb{R}^m & \rightarrow^{Id} & U_2 = U_1 \\
	\downarrow_{\theta} & &\downarrow_{\theta_1} & &\downarrow_{\iota}\\		\downarrow_{\theta} & &\downarrow_{\theta_1} & &\downarrow_{\iota}\\
	N\supset V_0 & \rightarrow^{\psi_0} & V_1\subset R^n & \leftarrow^{\xi} & V_2\\		N\supset V_0 & \rightarrow^{\psi_0} & V_1\subset \mathbb{R}^n & \leftarrow^{\xi} & V_2\\
	\end{matrix}		\end{matrix}
	</math>		</math>

Aptikuis: /* First Hour */

2007-10-06T01:23:49Z

First Hour

← Older revision		Revision as of 21:23, 5 October 2007
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	1) points ''pushforward'' <math>x\mapsto\theta_*(x) := \theta(x)</math>		1) points ''pushforward'' <math>x\mapsto\theta_*(x) := \theta(x)</math>

	2) Paths <math>\gamma:R\rightarrow X</math>, ie a bunch of points, ''pushforward'', <math>\gamma\rightarrow \theta_*(\gamma):=\theta\circ\gamma</math>		2) Paths <math>\gamma:\mathbb{R}\rightarrow X</math>, ie a bunch of points, ''pushforward'', <math>\gamma\rightarrow \theta_*(\gamma):=\theta\circ\gamma</math>

	3) Sets <math>B\subset Y</math> ''pullback'' via <math>B\rightarrow \theta^*(B):=\theta^{-1}(B)</math>		3) Sets <math>B\subset Y</math> ''pullback'' via <math>B\rightarrow \theta^*(B):=\theta^{-1}(B)</math>
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	<math>\theta_D_{\gamma}f = D_{\theta_\gamma}f</math>		<math>\theta_D_{\gamma}f = D_{\theta_\gamma}f</math>

	for functions <math>f:Y\rightarrow R</math>		for functions <math>f:Y\rightarrow \mathbb{R}</math>

	Now, <math>D_{\theta_\gamma}f = \frac{d}{dt}f\circ(\theta_\gamma)\|_{t=0} = \frac{d}{dt}f\circ(\theta\circ\gamma)\|_{t=0} = \frac{d}{dt}(f\circ\theta\gamma \|_{t=0} = D_{\gamma}(f\circ\theta) =\theta_*D_{\gamma}f</math>		Now, <math>D_{\theta_\gamma}f = \frac{d}{dt}f\circ(\theta_\gamma)\|_{t=0} = \frac{d}{dt}f\circ(\theta\circ\gamma)\|_{t=0} = \frac{d}{dt}(f\circ\theta)\circ\gamma \|_{t=0} = D_{\gamma}(f\circ\theta) =\theta_*D_{\gamma}f</math>

	''Q.E.D''		''Q.E.D''
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	Let us consider <math>\theta_*</math> on <math>T_pM</math> given a <math>\theta:M\rightarrow N</math>		Let us consider <math>\theta_*</math> on <math>T_pM</math> given a <math>\theta:M\rightarrow N</math>

	We can arrange for charts <math>\varphi</math> on a subset of M into <math>R^m</math> (with coordinates denoted <math>(x_1,\dots,x_m)</math>)and <math>\psi</math> on a subset of N into <math>R^n</math> (with coordinates denoted <math>(y_1,\dots,y_n)</math>)such that <math>\varphi(p) = 0</math> and <math>\psi(\theta(p)=0</math>		We can arrange for charts <math>\varphi</math> on a subset of M into <math>\mathbb{R}^m</math> (with coordinates denoted <math>(x_1,\dots,x_m)</math>)and <math>\psi</math> on a subset of N into <math>\mathbb{R}^n</math> (with coordinates denoted <math>(y_1,\dots,y_n)</math>)such that <math>\varphi(p) = 0</math> and <math>\psi(\theta(p))=0</math>

	Define <math>\theta^o = \psi\circ\theta\circ\varphi^{-1} = (\theta_1(x_1,\dots,x_m),\dots,\theta_n(x_1,\dots,x_m))</math>		Define <math>\theta^o = \psi\circ\theta\circ\varphi^{-1} = (\theta_1(x_1,\dots,x_m),\dots,\theta_n(x_1,\dots,x_m))</math>


	Now, for a <math>D\in T_pM</math> we can write <math>D=\~~sum_i a_~~{i=1}^m\frac{\partial}{\partial x_i}</math>		Now, for a <math>D\in T_pM</math> we can write <math>D=\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}</math>

	So, <math>(\theta_*D)(f) = \sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\varphi) = \sum_{i=1}^m a_i \sum_{j=1}^n\frac{\partial f}{\partial y_j}\frac{\partial\theta_j}{\partial x_i}=</math>		So, <math>(\theta_*D)(f) = \sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\varphi) = \sum_{i=1}^m a_i \sum_{j=1}^n\frac{\partial f}{\partial y_j}\frac{\partial\theta_j}{\partial x_i}=</math>
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	and so, <math>b_k = (\theta_*D)y_k =\begin{bmatrix}		and so, <math>b_k = (\theta_*D)y_k =\begin{bmatrix}
	0&\cdots & i & \cdots &0\\		0&\cdots & 1 & \cdots &0\\
	\end{bmatrix}		\end{bmatrix}
	\begin{bmatrix}		\begin{bmatrix}
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	\end{bmatrix}</math>		\end{bmatrix}</math>

	where the i is at the kth location.		where the 1 is at the kth location.