0
votes
0answers
28 views

Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
1
vote
0answers
64 views

A question about $1$-forms (on $S^3$)

What is meant by saying that any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a "co-closed" $1$-form? [...Since $H^1$ of $S^3$ is trivial it follows that the ...
10
votes
1answer
144 views

Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic ...
2
votes
1answer
530 views

What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and ...
2
votes
1answer
212 views

Spherical geometry - relating angles of lunes and segments of great circles

Consider the picture below. I have a sphere of radius $r$, centered at $C$. The angle $\varphi$ is the dihedral angle between the plane defined by the shaded area and a plane through the indicated ...
1
vote
2answers
102 views

Find an atlas of charts on $S^2$ with certain properties

Find an atlas of charts on $S^2$ for which each chart preserves area, and the transition functions relating charts have derivatives with determinant 1. I have been thinking that I should consider the ...
3
votes
1answer
369 views

Geodesics on a 2-sphere

I've been doing some work where I need to find the geodesics in a given Riemannian Manifold. Let's take the example of the two sphere, for simplicity, with unitary radius. The distance between two ...
1
vote
1answer
243 views

dotting gradient in spherical coordinates with displacement vector

The gradient in spherical coordinates is given by: $\nabla f = \left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial ...
1
vote
0answers
114 views

Stereographic projections and cross-ratios

Would anybody shed some light on question 2.11 in Wilson's Curved Spaces? The numbers $p,q\in \hat{\mathbb{C}}$ are stereographic projections of points $P,Q$ on the unit sphere. The spherical ...
0
votes
1answer
157 views

Geodesic segments on the unit sphere

Let $\mathbb{S}^2$ be the unit sphere and $d$ be the geodesic distance. For any three points $A,B,C\in \mathbb{S}^2$ and $0<\lambda<1$, let $A_{\lambda}$ and $B_{\lambda}$ be points on the ...
2
votes
1answer
2k views

Maximum sum of angles in triangle in sphere

Recently my differential geometry lecturer demonstrated that the sum of the interior angles of a triangle in a sphere is not necessarily never $180^\circ$. This is one way to prove that the earth is ...
4
votes
1answer
469 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...