## A metric has positive sectional curvature if and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$

This is a cross-post from my question on MSE.

It is well known that

In dimension three a metric has positive sectional curvature if and only if $${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$$. where $$r$$ denotes the scalar curvature.

Is there a higher dimensional analogues of this theorem? Any reference or counterexample?

## What are all pairs $(R,M)$ of a ring $R$ and a left $R$-module $M$ such that all endomorphisms of $M$ are scalar multiple of $\text{id}_M$?

I was playing with some endomorphism rings and got curious whether a classification of all (not necessarily unitary) module $$M$$ over a (not necessarily unital) ring $$R$$ such that for every $$R$$-module homomorphism $$\varphi:M\to M$$, there exists $$r\in R$$ such that $$\varphi(x)=r\cdot x$$ for all $$x\in M$$. I know that it means $$\text{End}_R(M)\cong R/\text{Ann}_R(M)$$ but I can’t make any interesting conclusion from this. In the case that $$R$$ is a division ring and $$M$$ is an $$R$$-vector space, then it is obvious that $$M$$ must be at most one-dimensional. Could you please give me some references if there are any studies regarding this question?

## “Almost” absolute continuity of derivative of BV function if ${\rm Tr}\,D_Sf = 0$

Let $$f: \mathbb R^N \to \mathbb R^N$$ be a $$BV$$ function. Suppose that $$\mathrm{div} f$$ is absolutely continuous with respect to the Lebesgue measure: $$\operatorname{div} f \ll \mathcal L^N$$. This implies, as seen in a related question, that $${\rm Tr}\,D_Sf = 0.$$

Does it mean that $$D_S f$$ is almost absolutely continuous in some sense? What is the correct way to formalize this notion of “almost” absolute continuity?

Here is a more precise question:

• As mentioned in the question Lusin Lipschitz approximation in BV and Sobolev space, $$f$$ is Lipschitz outside a small set (small with respect to the Lebesgue measure). Does $${\rm Tr} D_S f = 0$$ imply that this set is negligible with respect to the singular measure $$D_S f$$?

Related question are asked in the posts BV function with absolutely continuous divergence and Role of absolute continuity of divergence of BV function in proof of renormalization property

## Can’t grant myself permissions to execute rm command as sudo

I have a user account on my machine:

$(whoami) > foo_user  I’ve updated by /etc/sudoers file to grant myself permissions to use the rm command: User_Alias OPERATORS = foo_user Cmnd_Alias RM = /bin/rm OPERATORS ALL = RM OPERATORS ]ALL = (ALL) ALL  However, when I try to run rm on a directory, I get a permissions denied error: rm: /root-directory: Permission denied  What am I doing wrong? ## Disallow rm to remove * Note: If you googled this by the title of this question, don’t use this script unless you know what it is supposed to do. This is a script in bash 3+ that I have used for long for preventing rm * and rm -rf * from accidentally invoked and removing important files by mistake. I put it in my ~/.bash_aliases. alias rm='set -f;rm' rm(){ if [[ "$  -" == *i* ]]     then         if [ "$1" = "*" ] || [ "$  2" = "*" ] || [ "$1" = "./*" ] || [ "$  2" = "./*" ]         then             echo "Abort: refusing to remove *, please go to the parent folder and do rm <folder_name>/*" 1>&2             set +f             return 1         fi     fi       set +f     /bin/rm -i \$  @ }  set +f 

I would like to know whether there are any vulnerabilities and whether it can be improved.