# Tagged Questions

In the area of mathematics known as differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. (Def: http://en.m.wikipedia.org/wiki/Morse_theory)

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### Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
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### What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

While for $n=2$ it is pretty clear that the contours of a non-constant $f:\mathbb R^n\to\mathbb R$ are either extrema (and therefore points) or (the union of) 1-dimensional isolines, for $n=3$ I am ...
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### When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla f\right\rangle=\... 1answer 58 views ### Explicit verification of signs in Morse complex I'm trying to check by hand that the signs in the Morse complex, defined via choices of orientations on the unstable manifolds, lead to$\partial^2=0$. The books I've looked in seem to say either ... 1answer 72 views ### Relation between tangent spaces of (un)stable manifolds in Morse theory After asking this question about signs in the Morse complex, I realised that my confusion is really about how tangent spaces to different (un)stable manifolds are related. So suppose we have a Morse ... 1answer 128 views ### Fundamental theorem of Morse theory for$\Omega(S^n )$Using the Fundamental theorem of Morse Theory we can prove that$\Omega(S^n)$is homotopically equivalent to a CW complex with one cell each in dimensions$o,n-1,2(n-1), \cdots$and so on. But how can ... 1answer 83 views ### Path space of$S^n$Suppose that$p,q$are two non conjugate points on$S^n$($p \ne q,-p$). Then there are infinite geodesics$\gamma_0, \gamma_1, \cdots$from$p$to$q$. Let$\gamma_0$denote the short great circle ... 1answer 187 views ### The connection of Morse function Suppose$M$and$N$are two manifolds,$f$is a Morse function on$M$,$g$is a Morse function on$N$, can you find a new manifold$P$as the connection of$M$and$N$and a Morse function$h$on the$...
Let $f:M\to \Bbb{R}$ be a Morse function, where $M$ is a $k$-manifold. The index $i_{f,p}$ is defined to be the number of negative eigenvalues of the Hessian $H_f$ at the critical point $p$. For ...