All Questions
14 questions
11
votes
3
answers
1k
views
What is the minimum-curvature curve interpolating a given set of points in the plane?
We are given a set $X$ of $n\ge 3$ points in $\mathbb{R}^2$, belonging to the boundary of the convex hull of $X$ itself. Let $\Gamma(X)$ be the set of all convex, simple closed curves in $\mathbb{R}^2$...
- 1,677
0
votes
0
answers
252
views
Computation of scalar curvature from a Riemannian metric
I want to compute the scalar curvature for points on an empirical manifold (sampled data).
I have already an algorithm that learns the Riemannian metric and computes geodesics, so from the metric I ...
4
votes
1
answer
369
views
Comparing two Riemannian metrics on Grassmannian
Let $G_r(n)$ be the real Grassmannian which is the collection of all $r$ dimensional subspace in $\mathbb{R}^n$ equipped with the usual invariant metric $g$.
Let $U_A\in\mathbb{R}^{n\times r}$ and $...
- 1,495
9
votes
1
answer
529
views
Ricci Curvature on Grassmannian
Suppose $G_r(n)$ is the Grassmannian, which is the collection of all $r$ dimensional subspace in $\mathbb{R}^{n}$ equipped with the usual invariant metric. Let $Ricc(G_r(n))$ be the Ricci curvature ...
- 1,495
0
votes
1
answer
181
views
Convex planar curves and intersections [closed]
Given two planar regular convex not-closed curves C and C_1.
Let A the set of finite intersections between C and C-1.
Then what is the stricter upper bound of |A|?
I would say 2.
Thanks.
2
votes
1
answer
489
views
An identity for Futaki-Donaldson invariant
Let $(X,L)$ be a polarized projective variety
Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of :
a scheme $\mathfrak X$ with a $\mathbb C^*$-action
a ...
user21574
20
votes
5
answers
1k
views
Historical use of figures in geometry
I was surprised to learn from John Stillwell's comment in answer to the
question,
"Can the unsolvability of quintics be seen in the geometry of the icosahedron?",
that
There is not a single picture ...
- 151k
1
vote
0
answers
142
views
Relationship between stabilizers of a general point and a boundary point
Let $V$ be an n-dimensional complex vector space, and $u\in S^nV$ be a polynomial, $G(u)$ be the stabilizer of $u$ in $GL(V)$. Let $[v]\in\overline{GL(V)\cdot[u]}\subset\mathbb{P}(S^nV)$, but $v\notin ...
- 105
23
votes
12
answers
15k
views
Textbook for undergraduate course in geometry
I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (...
4
votes
0
answers
152
views
Is there an ellipsoid with given outer normals?
Pick two points $(x,0)$ and $(0,y)$ (say $x>0$ and $y>0$). Pick a unit vector $u = (u_1,u_2)$, $v = (v_1, v_2)$, and attach one to each of the points. Provided $u$ and $v$ are "nice" ($v$ needs ...
- 1,069
11
votes
2
answers
2k
views
Non-Kahler Complex manifolds
For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define ...
9
votes
2
answers
3k
views
An optimization problem for points on the sphere (master's dissertation)
First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a ...
- 3,612
0
votes
1
answer
2k
views
Categories of Geometry [closed]
I learn that Geometry has several categories/subfields from Wikipedia. But I am still not clear about the standards according to which they are classified.
It seems Euclidean Geometry, Affine ...
- 357
102
votes
6
answers
11k
views
Is there an analogue of curvature in algebraic geometry?
I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...
- 29.2k