Computations on differential geometry – Mathematica Stack Exchange

I have to do some really long computations of basic riemannian geometry. I am trying to use Mathematica to check them. This is the kind of computations I want to do. Let $M$ be a riemannian manifold and $f:Mrightarrow M$ be given by $f(x)=x$, then I want a code that tells me that
$$df_x(e_i)=e_i,$$
for $xin M$ and $e_iin T_xM$.
Also some of the computations involve and inner product. For exampe, if $f:Msubset (mathbb{R}^3,langle, rangle) rightarrow mathbb{R}$ is given by $f(x)=langle x,xrangle$ then I would like to have a code that tells me that
$$df_x(e_i)=2langle x,e_irangle, $$
for $xin M$ and $e_iin T_xM$.
I would be thankful for any help.

ag.algebraic geometry – Do representations of same dimension implies isomorphic closed orbits?

Let us recall this fact. Let $G$ be a semisimple algebraic group over $mathbb C$ and let $V,V’$ be two irreducible $G$-representations. We denote by $X,X’$ the unique closed $G$-orbits contained in $mathbb P V, mathbb P V’$ respectively. We know that if
$$
mathbb P V supset X cong X’ subset mathbb P V’
$$

as projective $G$-varieties, then $mathbb PV cong mathbb PV’$ as projective spaces. In particular, $dim V=dim V’$.

I want to understand the inverse direction: if I have two irreducible $G$-representations $W,W’$ of the same dimension, should I conclude that the closed $G$-orbits $Y subset mathbb P W, Y’ subset mathbb P W’$ are isomorphic as projective $G$-varieties?

ag.algebraic geometry – About condition for structure sheaf of a scheme being compact object in a category of sheaf of module over X

I found the condition for one direction :
Categorical interpretation of quasi-compact quasi-separated schemes

This article said that if $X$ is quasi compact and quasi separated, $mathscr{O}_X$ is a compact object in $Qcoh(X)$. A comment said that if the structure sheaf is compact( the global section functor preserves a colimit in $Qcoh(X)$), then $X$ is quasi compact.

Is there any result about quasi-separatedness?

dg.differential geometry – How is $nabla_XY$ calculated along $X$?

Let $X,Y:Mto TM$ be vector fields. I know the expression of the covariant derivative $nabla_XY|_p$ as a limit given its parallel transport on a curve along the direction of $X$.
$$nabla_XY|_p=lim_{tto 0}frac{Pi_{-tX}(Y_{phi^X(t)})-Y_p}{t}$$
the curve satisfying:
$$phi^X(0)=p ; ; ; ; ; (phi^X(0))’=X_p$$
so what curve is $phi^X(t)$ exactly? an integral “flow” curve of $X$ passing through $p$? a geodesic in the direction of $X$? or what?

ag.algebraic geometry – How to construct a sheaf on the infinitesimal site from a stratified module

Let $Xto S$ be a morphism of schemes.
Proposition 2.11 of the book “Notes on crystalline cohomology” by Berthelot and Ogus states that a stratified $mathcal{O}_X$-module $(E,{varepsilon_ncolon P^n_{X/S}otimes Eto Eotimes P^n_{X/S}}_n)$ gives rise to a crystal on the infinitesimal site $mathrm{Inf}_{X/S}$ (and vice versa).
However, their (sketch of) proof only treats infinitesimal thickenings $U→T $ for $Usubset X$ which locally admit a retraction.
How can we define the value $E(T)$ for an infinitesimal thickening which does not admit a retraction even locally?
Is this proposition really true?

ag.algebraic geometry – Why do we always need the Schwarz lemma when bounding the trace of a Kähler metric? [Cross-posted from MSE]

I posted this question on MSE, and while it has received some upvotes, it is not getting much attention. Perhaps it is more relevant here?

My undergraduate thesis topic is Kähler geometry. The general direction is something like the Calabi-Yau theorem or more adventurously some singular Calabi-Yau theorem, but this is not certain yet. One thing that I am noticing a lot of in my reading of Kähler geometry is that if we have two Kähler metrics $omega$, $eta$, then to get a bound of the form $$text{tr}_{omega}(eta) leq C$$ we need to use the Schwarz lemma — Essentially, we apply the maximum principle to some term like $$log text{tr}_{omega}(eta) – A varphi,$$ where $omega = eta + dd^c varphi$ and $A>0$ is large. This requires an assumption on the (Ricci/bisectional/holomorphic sectional) curvatures of $omega$, $eta$ (depending on which Laplacian one computes with).

I feel that I understand how to use the Schwarz lemma to get these estimates, but I want to ask why we have to use it (if we have to?).

This is prompted by studying singular metrics, for examples cone and cusp metrics: To formulate my question, let $D$ be a divisor in a compact Kähler manifold $M$, and for simplicity, assume that $D$ has simple normal crossings. A cone Kähler metric is a Kähler metric which is smooth on $M – D$ and is quasi-isometric to $$frac{i}{2} sum_{j=1}^k | z_j |^{2(1-beta_j)} dz_j wedge doverline{z}_j + frac{i}{2} sum_{j geq k+1} dz_j wedge doverline{z}_j.$$

A cusp Kähler metric is a smooth Kähler metric on $M-D$ which is quasi-isometric to $$frac{i}{2} sum_{j=1}^k | z_j |^{-2}| log | z_i |^2 |^2 dz_j wedge doverline{z}_j + frac{i}{2} sum_{j geq k+1} dz_j wedge doverline{z}_j.$$

From these descriptions, can one not see immediately that if $omega$ is cusp and $eta$ is cone, then $$text{tr}_{omega}(eta) leq C | z_i|^2 | log | z_i |^2|^2,$$ which would give $$text{tr}_{omega}(eta) leq C prod_j | sigma_j |^2 | log | sigma_j |^2 |^2,$$ if $sigma_j$ are the defining sections for the divisor $D$?

What initially came to my mind is a coordinate dependence problem, but this seems to contradict the fact that many calculations of this type involve normal coordinate calculations.

Sorry if this question is silly.

ag.algebraic geometry – Unirational subvarieties of moduli spaces of curves

Let $overline{M}_{g,n}$ be the Deligne-Mumford compactification of the moduli space $M_{g,n}$ of $n$-pointed genus $g$ smooth curves, and $Xsubsetoverline{M}_{g,n}$ a unirational variety intersecting $M_{g,n}$.

Is there an upper bound on the dimension of $X$ depending on $g,n$ (at least for $g$ big enough when $overline{M}_{g,n}$ is of general type)?

Thank you.

dg.differential geometry – Do exist constant curvature manifolds (hyperbolic or elliptic) with torsion?


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geometry – Distances between the center of escribed circle and acute triangle sides

Prove that, if $d_a$, $d_b$, $d_c$ are distances between the center of escribed circle and acute triangle sides($a,b,c$ accordingly), then
$d_a$$+$$d_b$$+$$d_c$ $=$ $R$+$r$

I didn’t do much, because i can not quite think of anything useful except somehow using this equations

$d_aa$ $+$ $d_bb$ $+$ $d_cc$ $=$$(a+b+c)$ $*$ $r$ $=$ $2S$$=$$2√(p*(p-a)*(p-b)*(p-c))$

$R$$=$$abc$$/$$4S$

ag.algebraic geometry – Number of representations of a given dimension

Let $G$ be a (semi)simple algebraic group over $mathbb C$ and let $d in mathbb Z_{>0}$ a fixed integer.

Let us suppose that there is an irreducible $G$-representation $V$ such that $dim V=d$. Can we count in general how many more irreducible $G$-representations of dimension $d$ there are?

For sure, given $V$, there is also its dual $V^vee$. In which case is there something else? I mean, like in the case if triality where $G=Spin_8(mathbb C)$ where there are three irreducible representations $V(omega_1),V(omega_3),V(omega_4)$ of dimension $d=8$.