User talk:Harold

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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:

  1. 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.
  2. [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:

[math]\text{YOUR MATH HERE}[/math]

To float to the right:

[math]\xymatrix{A \ar[r]^f & B}[/math]
Now you can write here and reference the diagram on the right
  • Lists and everything
    • Baby

Hope it helps Alec (talk) 22:15, 14 February 2017 (UTC)