Difference between revisions of "User talk:Harold"
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The setup is as following. Let <m>(M, g)</m> be a closed (i.e., compact and connected) smooth Riemannian manifold (without boundary), and suppose <m>f: M \to \mathbb{R}</m> is a smooth map satisfying the following properties: | The setup is as following. Let <m>(M, g)</m> be a closed (i.e., compact and connected) smooth Riemannian manifold (without boundary), and suppose <m>f: M \to \mathbb{R}</m> is a smooth map satisfying the following properties: | ||
# for each <m>x \in \mathrm{Crit}(f) := { p \in M : df_p = 0 }</m>, the Hessian <m>\mathrm{Hess}(f): T_pM \times T_pM \to \mathbb{R}</m> is non-degenerate. '''TODO Define the Hessian.''' | # for each <m>x \in \mathrm{Crit}(f) := { p \in M : df_p = 0 }</m>, the Hessian <m>\mathrm{Hess}(f): T_pM \times T_pM \to \mathbb{R}</m> is non-degenerate. '''TODO Define the Hessian.''' | ||
| − | # <m>f|_{\mathrm{Crit}(f)} | + | # <m>f|_{\mathrm{Crit}(f)}: \mathrm{Crit(f)} \to \mathbb{R}</m> is injective. |
== Caveat with xymatrix == | == Caveat with xymatrix == | ||
Latest revision as of 20:49, 15 February 2017
A collection of thoughts on Morse theory
I'm currently trying to figure out why in Morse homology, the degree of the attaching map of a certain [ilmath]n[/ilmath]-cell is somehow equivalent to the number of gradient flow lines. The setup is as following. Let [ilmath](M, g)[/ilmath] be a closed (i.e., compact and connected) smooth Riemannian manifold (without boundary), and suppose [ilmath]f: M \to \mathbb{R}[/ilmath] is a smooth map satisfying the following properties:
- for each [ilmath]x \in \mathrm{Crit}(f) := { p \in M : df_p = 0 }[/ilmath], the Hessian [ilmath]\mathrm{Hess}(f): T_pM \times T_pM \to \mathbb{R}[/ilmath] is non-degenerate. TODO Define the Hessian.
- [ilmath]f|_{\mathrm{Crit}(f)}: \mathrm{Crit(f)} \to \mathbb{R}[/ilmath] is injective.
Caveat with xymatrix
Hey, try this page:
See how you can scroll right? Alec (talk) 22:08, 14 February 2017 (UTC)
Some copy-and-paste-help
It's good to render diagrams in tables, if only because they look a bit sparse with the white background (unless they're huge), try these:
To float to the right:
- Lists and everything
- Baby
