multivariable calculus – How define the derivative of a mapp between two manifolds?

What shown below is a reference from the text Differential from by Victor Gullemin an Peter Haine. I point out that if you know the definitoin of Manifold and tangent space you can read only the part of the image beyond the theorem $4.2.10$.

enter image description here

So I ask effectively why the derivative of the map $g$ at the points of $X$ does not by the extension $tilde g$ any why moreover by a particular parametrization: indeed if $phi$ and $psi$ are two different local patches about $p$ then $dtilde g=d(gcircphi)big(phi^{-1}(p)big)$ and $dtilde g=d(gcircpsi)big(psi^{-1}(p)big)$ but by the chain rule
$$
dtilde g(p)=d(gcircphi)big(phi^{-1}(p)big)=d(gcircpsi)big(psi^{-1}(p)big)cdot d(psi^{-1}circphi)big(phi^{-1}(p)big)=dtilde g(p)cdot d(psi^{-1}circphi)big(phi^{-1}(p)big)
$$

and unfortunately I think that $big(phi^{-1}(p)big)neq 1$ generally. So could someone help me, please?

Approximation of the locally stable and unstable manifolds of the origin for a system

Calculate an approximation of the locally stable and unstable manifolds of the origin for the system

$$
dot{x} = 6x+8y+sin^2{x}-y^2 \
dot{y} = 8x-6y+x^2+xy
$$

Attempt

I tried calculating the approximations using the Taylor expansion for the functions $h_sleft(xright),h_uleft(yright)$ where $h_sleft(0right) = 0 = h_uleft(0right)$ and
$frac{dh_s}{dx}left(0right) = 0 =frac{dh_u}{dy}left(0right)$

Using the equations

$dot{x}frac{dh_sleft(xright)}{dx} = dot{y}$
$dot{x}frac{dh_uleft(yright)}{dy} = dot{x}$

I approximated $h_sleft(xright)$, though I am not sure my calculations were correct because I wasn’t sure about how to handle $h_s^2left(xright)$ and $sin^2{left(xright)}$ as Taylor series expansions. I figured that if I initially had terms of up to $O(x^3)$ then whenever I came across a multiplication which resulted in higher power, I would disregard it (e.g. $x^2*x^3 = x^5 < O(x^3) Rightarrow x^5 approx 0 $ since we are interested in the case where $x rightarrow 0$.

Then I did the same for $h_uleft(yright)$ but then I would get a term $sin^2{left(h_uleft(yright)right)}$ which I am not sure how to handle as a Taylor expansion.

Does someone have any tip?

ag.algebraic geometry – Descendent Gromov-Witten invariants and Frobenius manifolds

I’ve heard it said that genus $0$ descendent Gromov-Witten invariants of a smooth projective variety $X$ can be encoded in the structure of a Frobenius manifold on the cohomology $H^*(X,mathbb{C)}$. I am very confused by this – it is clear that this is true for non-descendent Gromov-Witten invariants. How do the descendent invariants enter in?

dg.differential geometry – Does higher integrability of Jacobians hold between manifolds when the Jacobians are concentrated?

$newcommand{M}{mathcal{M}}$
$newcommand{N}{mathcal{N}}$

Let $M,N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds.

Let $f_n rightharpoonup f$ in $W^{1,2}(M,N) $ with $Jf_n > 0$ a.e., and suppose that the volume $V({x in M , | , Jf_n le r}) to 0$ when $n to infty$, for some $0<r<1$. Is it true that $ Jf_n rightharpoonup Jf $ in $L^1(M)$?

I am fine with assuming that $f_n$ are Lipschits and injective and that $V(f_n(M)) to V(N) $.


The “higher integrability property of determinants” implies that if $M,N$ are open Euclidean domains, then $ Jf_n rightharpoonup Jf $ in $L^1(K)$ for any compact $K subset subset M$.


Without the assumption $V(Jf_n le r) to 0$, this clearly doesn’t hold, even when $f_n$ are conformal diffeomorphisms:

Take $M=N=mathbb{S}^2$. Let $s: mathbb{S}^2 to mathbb{R}^2 cup {infty}$ be the stereographic projection, and let $g_k(x) = k x$ for $x in R^2$ (and $g_n(infty) = infty$.).

Set $ f_n = s^{-1} circ g_n circ s$. $f_k$ are conformal, orientation preserving, smooth diffeomorphisms
and thus $ int_{mathbb{S}^2 }Jf_n=V(mathbb{S}^2 )$. By conformality $int_{mathbb{S}^2 } |Df_n|^2 =2int_{mathbb{S}^2 }Jf_n$ is uniformly bounded, so $f_n$ is bounded in $W^{1,2}$, and converges to a constant function. (asymptotically we squeeze bigger and bigger parts of the sphere to a small region around the pole).

So, we do not have weak convergence of $Jf_n$ to $Jf=0$. (the
Jacobians converge as measures to a Dirac mass at the pole.) The question is if by adding the non-degeneracy constraint $V(Jf_n le r) to 0$ we recover this ‘Jacobian Rigidity’ under weak convergence.


*(In my case of application $r=frac{1}{4}$ but I don’t think it matters).

dg.differential geometry – Regarding projective manifolds with decomposable real tangent bundle

Let $X$ be a complex projective manifold. Suppose its real tangent bundle $T_{mathbb{R}}X$ splits as a direct sum of two sub-bundles of even rank. Does this give any useful information about the manifold $X$ (e.g. its Euler characteristic, etc.)?

I apologize for being somewhat vague about my question. Any help would be appreciable.

manifolds – about Cartan’s theorem

theorem : Let $SL(n,Bbb{C})$ be the group of matrices of complex entries and determinant $1$ then $SL(n,Bbb{C})$ is a regular submanifold of $GL(n,Bbb{C})$.

proof : $GL(n,mathbb{C})$ is a Lie group and $SL(n,mathbb{C})$ a closed subgroup. with Cartan’s theorem $SL(n,mathbb{C})$ is a regular Lie subgroup, in particular a regular submanifold.

Cartan’s theorem : if $H$ is a closed subgroup of a Lie group G, then H is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding.

why $SL(n,mathbb{C})$ is a closed subgroup ? what is dimension of manifold $SL(n,mathbb{C})$ ?

gt.geometric topology – $0$-surgery of slice knots and contractible manifolds

We know that if we attach $4$-dimensional $2$-handle $D^2 times D^2$ to $S^1 times S^2$, then we produce a contractible $4$-manifold. In this case, $S^1 times S^2$ is $0$-surgery on the unknot.

If we replace the unknot with a slice knot, can we still have a contractible manifold? Is there easy way to argue this?

Note: Here, slice knots bound smoothly properly embedded disks in $4$-balls.

smooth manifolds – Realising the non-trivial orientable $S^2$-bundle over $T^2$ as a quotient of $S^2times T^2$

Using the fact that $operatorname{Diff}^+(S^2)$ deformation retracts onto $SO(3)$, one can show that over any connected, closed, smooth surface, there are two orientable $S^2$-bundles, the trivial one and a non-trivial one. Moreover, the two bundles can be distinguished by their second Stiefel-Whitney class.

Let $X$ be the total space of the non-trivial orientable $S^2$-bundle over $T^2$. If we pullback $X to T^2$ by any double covering $T^2 to T^2$, we obtain the trivial $S^2$-bundle $S^2times T^2 to T^2$. It follows that $X$ is double covered by $S^2times T^2$.

What is the free $mathbb{Z}_2$ action on $S^2times T^2$ which has quotient $X$?

It would be enough to realise the total space of a rank two bundle $E to T^2$ with $w_2(E) neq 0$ as a $mathbb{Z}_2$ quotient of $T^2timesmathbb{R}^2$.

Note that $X$ can be realised as the sphere bundle of $gamma_1oplusgamma_2oplus(gamma_1otimesgamma_2)$ where $gamma_i$ is the pullback of the non-trivial line bundle on $S^1$ by projection onto the $i^{text{th}}$ factor. It is easy to see pulling back by $(z, w) mapsto (z^2, w)$ (or $(z, w) to (z, w^2)$), the bundle becomes trivial (as it must). I was hoping these explicit descriptions would help, but so far I have been unsuccessful.

manifolds – can $(x^3 , y)$ be a chart if $(x , y)$ is a chart on $M$?

Let $M$ be a $2$-dimensional manifold and $(U , varphi)$ be a chart on $M$ with $varphi = (x , y)$. Can $(U , psi)$, with $psi = (x^3 , y)$, be a chart on $M$? I thought no because if it were a chart, the map $psi = g circ varphi : U to {mathbb{R}}^2$ should be a diffeomorphism, with $g : {mathbb{R}}^2 to {mathbb{R}}^2$ given by $g(a , b) = (a^3 , b)$. But it implies that $g$ is also a diffeomorphism, but the first partial derivative of $g^{- 1} : (a , b) mapsto (a^{frac13} , b)$ does not exist.

My argument should be wrong, as on can be read on $textbf{Lee’s Smooth Manifold}$ page 65, that $(tilde{x},tilde{y})$ in $mathbb{R}^2$ related by
$$
tilde{x} = x, qquad tilde{y} = y+x^3
$$

is a chart if $(x , y)$ denotes the standard coordinates on ${mathbb{R}}^2$. Where is my mistake?

manifolds – On the dimension of the projected rank $r$ matrix space

In $d$-dimensional complex number space $mathbb{C}^{ d}$, we can define the rank at most $r$ matrix space
$$
S={A| mathrm{rank}(A)leq r}subseteq mathbb{C}^{dtimes d}.
$$

The dimension of manifold $S$ is roughly $rd$.

Suppose we are given a basis of $mathbb{C}^{dtimes d}$, says $A_1,A_2,…,A_{d^2}$, then, any $Ain S$ can be written as the linear combination of $A_1,A_2,…,A_{d^2}$.

We choose $A_1,…A_k$ and compute the projection of $S$ onto the span of $A_1,…A_k$, denoted as
$$
T={B| B=sum_{i=1}^k alpha_i A_i, s.t, C=B+sum_{j=k+1}^{d^2} beta_j A_jin S mathrm{for} mathrm{some} alpha_i,beta_jin mathbb{C}}.
$$

What is the dimension of $S$?
What is the size of the $epsilon$-net of $T$ when we only consider elements in $T$ with 2 norm less than 1, for $epsilon>0$ and 2 norm?