5 votes
Accepted

Why can we use fundamental vector fields for vertical vector fields to prove the form of the curvature of a connection?

Lemma: Let $M$ be a smooth manifold, $\omega \in \Omega(M)$ be a differential one-form on M, and $\text{d}\omega:\Gamma(TM)\times\Gamma(TM)\rightarrow C^{\infty}(M)$ be the exterior derivative of $\...
  • 66
5 votes
Accepted

Does there exist a smooth structure on $[0,\infty)$ that induces the standard topology?

No. Assume there exists a smooth structure on $M = [0,\infty)$ that induces the standard topology. Then $M$ would be an $n$-dimensional smooth manifold for some $n$. In particular $M$ would be locally ...
5 votes
Accepted

Why p-linear alternating form evaluated on linearly dependent set of vectors is zero?

Assume that $\{v_1,\ldots,v_p\}$ is linearly dependant, and without loss of generality, assume that $v_p = \sum_{j=1}^{p-1} \lambda_j v_j$. Let $\omega$ be any $p$-alternate form. Then by ...
  • 11.3k
3 votes

existence of two different atlases in $S^{7} \subseteq \mathbb{R}^8$ that are not diffeomorphic

Just to be clear, this is in no sense an "exercise" unless you were specifically fed a bunch of results leading up to it. Exotic $7$-spheres were first constructed by Milnor in his 1956 ...
3 votes

Is the inverse function theorem for manifolds applicable here?

In fact, if $D\varphi(x)$ has maximal rank, then $\varphi(x)$ cannot lie in the boundary of $M$. Consider $d\!M$ the double of $M$, which consists in two copies of $M$ glued together along their ...
  • 11.3k
3 votes
Accepted

Question about covariant derivative on manifolds.

Presumably, the author means that $\nabla^i$ is the pullback through $\varphi_i$ of the covariant derivative $\bar \nabla$ on $\Bbb R^n$ defined by taking the directional derivatives in the following ...
  • 11.3k
3 votes
Accepted

Differentiating curves given by curves in a Lie group acting on a point

I think this should work out well if you use the appropriate logarithmic derivative (which I think is the right logarithmic derivative in this case). Let $\ell:G\times V\to V$ be the smooth map ...
  • 17.9k
2 votes
Accepted

Change of basis of tangent space Lee, Smooth Manifolds

You are right, if we write a matrix $A$ in the form $A = (a_{ij})$, then the first index $i$ denotes the row number in which we find the entry $a_{ij}$ and the second index $j$ denotes the column ...
  • 61.3k
2 votes
Accepted

Hessian of a function on a Riemannian manifold.

The Leibniz rule says that if $\alpha$ is a $1$-form and $X$, $Y$ are vector fields, then $X(\alpha(Y)) = (\nabla_X\alpha)(Y) + \alpha(\nabla_XY)$. (Side note: It is also true with Lie derivatives, ...
  • 11.3k
2 votes

Is real projective space a manifold with boundary?

Remember that a manifold with boundary is a topological space $M$ which is locally homeomorphic to either $\mathbb R^n$ or the half-space $\mathbb H^n:=\{(x_1,\dots,x_n)\in\mathbb R^n:x_1\ge0\}$. Then,...
  • 10.9k
2 votes

How is the harmonic map related to the second fundamental form?

Here are some references where the authors explain this fact: Fanghua Lin & Changyou Wang's The Analysis of Harmonic Maps and Their Heat Flows (pages 2 and 3) Struwe's chapter on Nonlinear ...
  • 1,559
1 vote
Accepted

Do Carmo Riemannian geometry Christoffel symbols

In general, if $A = (a_{ij})$ is a matrix, and $A^{-1}= (a^{ij})$ denotes it's inverse, then we can compute each entry of $E_n = A\cdot A^{-1}$ as follows: $$\sum_k a_{ik} a^{kj} = \delta_{i,j},$$ ...
1 vote
Accepted

Rotation of a Curve

In their proof of the Fundamental Theorem of Curve Theory the authors of [1] start with the following second order ODE for the tangent vector $\mathbf{t}$ w.r.t. the natural parametrization $s$: $$\...
  • 5,155
1 vote

Can I build fibre bundles over bundles ad infinitum?

Of course, you can iterate the tangent bundle construction infinitely and obtain "new" bundles in this way. The question of when it stops being "interesting" from the mathematical ...
  • 1,261
1 vote

Example for which the Donaldson-Futaki invariant is $0$

I can comment on the case when $X$ is smooth. In this case we are looking at an anti-canonically polarized (hence Kähler) manifold, and this is the setting in which K-stability was first studied. ...
1 vote

If two charts are smoothly compatible with the atlas, then they are naturally compatible

Let $(U_1,\varphi_1)$ and $(U_2,\varphi_2)$ be two charts compatible with some atlas $\mathcal{A}=\{(W_{\alpha},\psi_{\alpha})\}_{\alpha}$. Let $U:=U_1\cap U_2$. Case 1. Suppose that $U = \varnothing$....
  • 11.3k
1 vote

Rotation of a Curve

The claim is false. For example, let us consider $\underline{c}=(-1,0)$ and $\gamma(t)=(t,0)$ (a line): then $$\underline{t}(s)+\underline c=0$$ and so the curve obtained by integrating is actually a ...
1 vote
Accepted

When are the transition functions of a smooth complex vector bundle homotopic to a constant map?

If $U$ is not simply connected, then the transition function $\tau:U\to\text{GL}(n,\mathbb{C})$ will typically not be homotopic to the constant identity map. So no, you cannot guarantee this. ...
  • 1,261
1 vote
Accepted

existence of two different atlases in $S^{7} \subseteq \mathbb{R}^8$ that are not diffeomorphic

There's $28$ $S^7$'s and they form an abelian group. John Milnor discovered them, and probably won a Field's medal for it. They're homeomorphic but not diffeomorphic. In any dimension other than $4$...
  • 4,111
1 vote

Do Carmo Differential Geometry book: Proposition 4, Chapter 2.2. Mistake in the proof?

This is my attempt at building on Tim Bratten's response to show that $\mathbf{x}(V_1)$ is necessarily open. By Proposition 3 of Chapter 2.2, for each $q' \in V_1$, we can find a neighborhood $W$ in $...
1 vote

Vector Field on the Real Projective Plane F-related to Vector Field on Plane

The action of the Jacobian is not the same as above, I believe that the matrix takes the basis $\left\{\frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right\}$ (induced by $\varphi_3$) to the ...
1 vote

Proof that a differentiable manifold has a countable atlas verifying certain property

There is nothing to prove. By a chart you mean any member of the differentiable structure (maximal atlas) $\mathfrak D$ of $M$. You construct a countable subset $\mathcal A = \{(U_j,\varphi_j)\}_{j\in ...
1 vote
Accepted

Given a pair $(\zeta^2,v)$ find a pair $(\zeta^3,w)$ where there's an embedding $e:\zeta^2 \hookrightarrow \zeta^3$ s.t. $v=e^*w$

$e(x,y)=(x,y,0)$ (possibly $ (x,y,1))$ and define $$ W(x,y,z)=\bigg(x{\rm log}x,-y{\rm log}y,-z{\rm log}z \bigg) $$ Then $de\ \frac{\partial }{\partial x} = (1,0,0),\ de\ \frac{\partial }{\partial y}...
  • 19.3k
1 vote

Example of "giving" a metric to an intrinsically defined manifold

Here is a nice example: complex projective space $\mathbb{CP}^n$ is a space parameterizing $1$-dimensional complex subspaces of $\mathbb{C}^{n+1}$. It turns out to be a manifold, and it can be ...
1 vote
Accepted

Simple computation on the identity on the definition of Donaldson-Futaki invariant

It seems there is a mistake in the degrees of the polynomials, in your computation. Notice that $w_k$ is of degree $n+1$, while $d_k$ is of degree $d$. So, the two polynomials in the numerator and ...
1 vote

About closed curves in a dynamical systems

Here's a sketch for a solution that might work. Look at a simpler case first: that is for a circle $S^1$. Suppose we have a $C^1$ flow defined on $S^1$. We can express this as an ODE using polar ...
1 vote
Accepted

Differential structure on $\mathbb S^n/±\mathrm{id}$ such that the injection $i:\mathbb RP^n\rightarrow\mathbb S^n/(±\mathrm{id}$ is a diffeomorphism

As you say, the the natural injection $i:\mathbb S^n\rightarrow\mathbb R^{n+1}\backslash\{0\}$ induces a continuous map $i': \mathbb S^n/±\mathrm{id} \to \mathbb RP^n$ on the quotient spaces. That $i'$...
  • 61.3k

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