All Questions
11 questions
9
votes
3
answers
696
views
I want to find a smooth section of the map from the Stiefel manifold to the Grassmanian manifold
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 155
3
votes
1
answer
331
views
Variant of Wahba's problem
Wahba's problem is the following:
$$\min_R \sum_{k=1}^K \|v_k - Rw_k\|^2$$
where $v_k$ and $w_k$ are arbitrary $3\times 1$ vectors, and $R$ is a rotation matrix (i.e., orthogonal with $\det(R)=1$).
A ...
6
votes
0
answers
353
views
Atiyah–Singer Index theorem for the pedestrian / layperson
So I came across the so-called Atiyah–Singer Index Theorem (ASIT) and claims of it being an extremely powerful and versatile tool.
Question. What is a truly simple application of the ASIT to obtain a ...
- 6,853
32
votes
2
answers
1k
views
A question about subspace in ${\bigwedge}^2({\mathbb R}^n)$
Let $E$ be a linear subspace of ${\bigwedge}^2({\mathbb R}^n)$. What is the minimal dimension of $E$ that guarantees $E$ contains a nonzero element of the form $X\wedge Y$, with $X, Y\in{\mathbb R}^n$?...
- 637
3
votes
0
answers
131
views
What subdomains of $\mathbb{R}^2$ are diffeomorphic to $\mathbb{R}^2_+$ via rational functions?
For what subdomains $D \subset \mathbb{R}^2$ does there exist a rational diffeomorphism $h:D \to \mathbb{R}^2_+$, such that its inverse is also a rational function? (By "rational function" I mean a ...
- 111
7
votes
2
answers
315
views
Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices
I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...
- 6,206
5
votes
0
answers
604
views
The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)
(Migrated from MSE)
While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
- 10.5k
5
votes
2
answers
1k
views
How to calculate the determinant bundle
Maybe, this is a problem of linear algebra. But I do not know how to calculate it. Let $E$ be a vector bundle of rank $2$ over an algebraic surface. If $H=S^{2n}E\bigotimes (\operatorname{det} E)^{\...
- 713
-3
votes
1
answer
375
views
Opposite complex structure on Kaehler manifold
Let $(M,J)$ be a Kaehler manifold. How can one describe the opposite complex structure? What is the precise definition of the opposite complex structure? Can one describe the opposite complex ...
- 11
1
vote
1
answer
1k
views
General Orthogonal Group and its properties
I know that exist a Lie Group Called the Orthogonal Group $O(n)$.
That correspond to all matrix of $n \times n$ in the real numbers such that the columns are a orthogonal basis for $\mathbb{R}^n$. Is ...
- 335
6
votes
3
answers
2k
views
Finding the action of the symplectic group on the Siegel-half plane
Let $S$ be the Siegel-half plane of dimension $n$, i.e. the set of complex $n \times n$ matrices $Z$ which are symmetric and whose imaginary part is positive-definite. In dimension 1 we can identify $...
- 4,343