3 votes

In which points is $f: \mathbb{R}P^2 \rightarrow \mathbb{R}^3$ an inmersion.

Your chart $\phi_1$ is the one you would use if working with the projective space as the quotient $\Bbb{R}^3\setminus\{0\}\to\Bbb{RP}^2$, not $S^2\to\Bbb{RP}^2$. Thus your whole calculation is wrong. ...
peek-a-boo's user avatar
1 vote

Extending Diffeomorphisms of Lie Manifolds

Not assuming $N$ is connected: Let $H \cong S^1 \times \mathbb Z_3$ which topologically is the disjoint union of $3$ circles embedded as $H$ concentrically into $M = G = \mathbb R^2$ in such a way ...
Sven-Ole Behrend's user avatar
1 vote
Accepted

Structure required for differentiability in topological context

In order to talk about differentiability you need the full structure of $\mathbb{R}^n$, i.e. its vector space structure together with the Euclidean norm. Because these are essential in defining ...
freakish's user avatar
  • 43.2k
1 vote
Accepted

Reflection of the "Figure Eight" not diffeomorphic to the "Figure Eight"

A look into Boothby's book shows that he uses the concept of an imnmersed submanifold (see Definition 4.3). $N = G( \mathbb R)$ is regarded as such an object. This means that $N$ is not regarded as a ...
Paul Frost's user avatar
  • 76.6k
1 vote

Integral formula for Hopf map in local coordinates appears to vanish

This is obvious to me now that I've taken a step back to think about it! The form $\omega$ is itself a pullback, $\omega = f^*(-\cos\theta\,\mathrm{d}\phi)$, so $$ \omega\wedge\mathrm{d}\omega=f^*(-\...
xzd209's user avatar
  • 335
1 vote
Accepted

Immersions are local diffeomorphisms

Recall the definition of differentiability on manifolds. A map $f:M \rightarrow N$ between differentiable manifolds is called differentiable if, for all compatible charts $(U,\phi) \in A_M$, (V,$\psi)...
Lourenco Entrudo's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible