I am looking for simple (but not worn-out) application of Bochner--Weitzenböck type formulas in comparison geometry. (I want to use it as a motivation for students.) The vanishing theorems and estimates for eigenvalues are too standard. One of my favorite examples is the result of Fengbo Hang a...

I am looking for the reference to the following theorem. I have to apply a similar statement, and it would be nice to trace the source. Please note, I know few proofs in fact it is Problem 3 in my collection of exercises. Theorem. Let $\gamma$ be a closed simple plane curve with curvature ...

Let $B_{\infty}$ denote the infinite strand braid group. Let $\mathrm{sh}:B_{\infty}\rightarrow B_{\infty}$ be the group homomorphism where $\mathrm{sh}(\sigma_{i})=\sigma_{i+1}$ for all $i>0$. Define an operation $*$ on $B_{\infty}$ where $$x*y=x\cdot\mathrm{sh}(y)\cdot\sigma_{1}\cdot\mathrm{sh...

Let $B_{\infty}$ denote the infinite strand braid group. Let $\mathrm{sh}:B_{\infty}\rightarrow B_{\infty}$ be the mapping where $\mathrm{sh}(\sigma_{i})=\sigma_{i+1}$ whenever $i\geq 1$. Then $B_{\infty}$ can be endowed with an operation $*$ known as shifted conjugacy where $*$ is defined by $$x...