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 […]

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 […]

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

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 $\varphi_{b−a}$ […]

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 […]

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) = […]

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 […]

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) […]

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