A basic proof on Morse Function

Question

The questions is to show if $f_t$ is a homotopic family of functions on $R^k$, show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for sufficiently small t.

I get the previous question leads to the proof, but couldn't reach the question:

det$(H)^2 + \sum_{i=1}^k (\partial f/ \partial x_i)^2 > 0$

And also, on a compact manifold:

det$(H)^2 + \sum_{i=1}^k (\partial f/ \partial x_i)^2 >\epsilon$

Thanks for your help!

What is $H$? What is $\epsilon$? Can you provide a little more context, please? — Jesse Madnick, May 26 '13 at 00:52
@JesseMadnick Dear Jesse, Thank you so much! H is the Hessian Matrix, and $\epsilon$ just means an arbitrarily small positive number. Thank you! — 1LiterTears, May 26 '13 at 00:55

score 1 · Accepted Answer · answered May 26 '13 at 01:39

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G&P have you prove that the first inequality is equivalent to $f$ Morse. If the quantity is $>\epsilon$ for $f_0$, then by smoothness, for small enough $t$ it'll be $>\epsilon/2$ for $f_t$.

answered May 26 '13 at 01:39

Ted Shifrin

115,160

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Thank you Ted. I don't understand why it'll be $>\epsilon/2$ for $f_t$. Thank you very much for your help. – 1LiterTears May 26 '13 at 01:43
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What fundamental facts do you know about continuous functions? – Ted Shifrin May 26 '13 at 01:44
Continuous function has max/min on a compact set? – 1LiterTears May 26 '13 at 01:48
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And how does that minimum value vary if you wiggle $t$? – Ted Shifrin May 26 '13 at 01:53
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If $t$ is wiggled, $H_t$ and $\partial f_t/ \partial x_i$ change. So I get confused. – 1LiterTears May 26 '13 at 02:30
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What's the definition of continuity in your topology courses? – Ted Shifrin May 26 '13 at 02:35
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A function f : X→ Y between topological spaces is called continuous if for all x ∈ X and all neighbourhoods N of f(x) there is a neighbourhood M of x such that f(M) ⊆ N. – 1LiterTears May 26 '13 at 03:02
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OK, you need to think now about how this all fits together. I've given enough hints. Where are you a student? – Ted Shifrin May 26 '13 at 03:10
Thanks - let me sleep with it and see if I can figure it out. – 1LiterTears May 26 '13 at 03:57
Oh - I just graduated this year. =) – 1LiterTears May 26 '13 at 03:59
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So where are you studying differential topology? A key tool throughout this course (and elsewhere in mathematics) is the so-called Tube Lemma: If $X$ is compact and $U$ is a neighborhood of $X\times {y_0}$ in $X\times Y$, then $U$ contains a "tube" $X\times V$, for some neighborhood $V$ of $y_0$ in $Y$. – Ted Shifrin May 26 '13 at 19:19
Hi Ted, thanks for inform me that! I am reading GP by myself. :-) – 1LiterTears May 26 '13 at 20:49

A basic proof on Morse Function

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