Articles of morse theory

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 that this sign verification is easy (e.g. Audin-Damian), or prove $\partial^2=0$ by introducing more sophisticated notions like orientations of determinant line […]

Proof of Reeb's theorem without using Morse Lemma

I’m trying to prove Reeb’s theorem as stated in Milnor’s Morse Theory. That is, suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both non-degenerate), then $M$ is homeomorphic to $S^n$. I understand the most crucial part. That is, let $f$ be normalized such that the critical […]

Getting rid of square root via integration

How do we prove, for positive $D$, this result? $$ e^{-2\sqrt D} \sqrt{\pi} = \int_0^\infty s^{-1/2} e^{-(s+D/s)} ds $$

An other question about Theorem 3.1 from Morse theory by Milnor

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 $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla f\right\rangle=\dfrac{\mathrm{d}(f\circ c)}{\mathrm{d}t}$ I have three question please: 1) What is the purpose of seeing that $t\rightarrow f(\varphi_t(q))$ is linear with derivative +1 ? 2)How to see that $\varphi_{b−a}$ […]

Smooth classification of vector bundles

I am trying to understand the smooth classification of $n$-disk bundles over $S^n$. As vector bundles, these are classified by $\pi_{n-1}(SO(n))$ via the clutching construction but I am interested in their smooth type. For example, the map $\pi_{n-1}(SO(n)) \rightarrow \pi_{n-1}(Diff(D^n))$ might not be injective in which case two distinct vector bundles would be the same […]

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 the trajectories of the gradient flow $x'(t) = – \operatorname{grad} f(x(t))$ to analyse the space. In particular the (un)stable manifolds $$W^\pm(p) = […]

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 already not sure whether besides points and 2-dimensional isosurfaces there is also the possibility of (1D) isolines or not. I tried starting with […]

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, $x_1,x_2\in\mathbb{R}$ and $f(x_1)=f(x_2) = 0$. (This part is easy). ii) Show that there does not exist a smooth function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$, s.t. $f(x,y) \geq 0$, $\forall (x,y) […]