Tagged Questions
4
votes
0answers
68 views
Solution to $\Delta_g u = \delta-1$ on a 2-sphere.
Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
5
votes
1answer
55 views
Zeta Regularized Determinant of Laplacian
Can anyone point me to a resource where the zeta regularized determinant of the Laplacian is explicitly computed for simple two dimensional surfaces, say a rectangle or torus or cylinder?
6
votes
2answers
166 views
Elliptic estimates on compact manifolds
Hey where may I find elliptic estimates for PDEs on compact (no boundary) Riemannian manifolds? I want a source/paper/book where I can cite it.
For example, for $L$ a linear elliptic operator, (eg. ...
4
votes
1answer
73 views
Manifold without conjugate points and positive curvature
I'm looking for an example of a complete riemannian manifold with sectional positive curvature and without conjugate points. I've tried the projective space, but the identfication used to construct it ...
3
votes
2answers
151 views
Riemannian Geometry book to complement General Relativity course?
What would be a good Riemannian Geometry (or Differential Geometry) book that would go well with a General Relativity class (offered by a physics department)? I'm in one right now, but I'd like a pure ...
3
votes
1answer
60 views
What are all isometry classes of the 2-sphere?
In topology, one learns how to classify the compact surfaces up to homeomorphism. And in fact, since "homeomorphic" and "diffeomorphic" coincide in dimension 2, we can classify the compact (smooth) ...
2
votes
1answer
206 views
control of the $C^{1}$ norm of a diffeomorphism
Let $\mathcal{E}$ be the set of smooth manifolds with boundary $E\subset \mathbb{R}^{3}$ which are perturbations of the unit ball whose volume $V$, diameter $d$ and area of the boundary $A$ satisfy:
...
2
votes
1answer
123 views
good problem book in differential geometry
What are the books in Differential Geometry with good collection of problems. At present I am having John M.Lee's Riemannian Manifolds,Kobayashi Nomizu's Foundations of Differential
Geometry. I ...
1
vote
1answer
186 views
Weitzenböck Identities
The Wikipedia page for Weitzenböck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In ...
4
votes
2answers
154 views
Moving to a conformal metric
Given a generic 2-dimensional metric
$$
ds^2=E(x,y)dx^2+2F(x,y)dxdy+G(x,y)dy^2
$$
what is the change of coordinates that move it into the conformal form
$$
...
5
votes
2answers
245 views
Learning differential/Riemannian geometry for PDEs
I know there have been threads on which books to learn DG/RG from but hopefully this is sufficiently different to avoid closure.
Can anyone recommend a book to learn DG/RG (whichever is appropriate) ...
6
votes
1answer
114 views
Do elliptic operators on Riemannian manifolds have a regularizing effect?
I'm working on my master thesis and need to handle some spectral theory of the Laplace operator on compact Riemannian manifolds and especially on the sphere. While investigating essential ...
8
votes
2answers
191 views
Cartan Theorem.
Cartan Theorem: Let $M$ be a compact riemannian manifold. Let $\pi_1(M)$ be the set of all the classes of free homotopy of $M.$ Then in each non trival class there is a closed geodesic. (i.e a closed ...
0
votes
0answers
59 views
Are there relevant non-complete riemannian manifolds?
Almost all theorems on riemannian geometry I know so far talk about complete riemannian manifolds. This hypothesis is useful either in positive (Bonnet-Myers theorem, for instance) or negative ...
7
votes
1answer
108 views
A scalar product in the space of oriented volumes?
Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
5
votes
3answers
322 views
Isometries of $\mathbb{S}^n$
How to prove so elementary (elementary = without using the concept of geodesic) that an isometry of $\mathbb{S}^n$ is a restriction on $\mathbb{S}^n$ of an isometry of $\mathbb{R}^{n+1}$ ?
EDIT:
...
4
votes
0answers
134 views
Riemannian Connection (Very basic question)
We know that a connection $\nabla$ in a manifold M hashas the purpose of performing the same role as the covariant derivative of vector fields of surfaces in $\mathbb{R}^3$. Such analogies are ...
1
vote
1answer
168 views
Need references on Cartan's method of moving frames.
Could anyone suggest a book or a paper containing a good, modern treatment to the Cartan's method of moving frames. Especially, I am interested in its use in studying geometric properties of surfaces ...
3
votes
1answer
157 views
Thrice-punctured sphere
This claim is made in the book Quantum Triangulations (eds.: Carfora, Marzuoli), p.45:
the thrice-punctured sphere is the largest subdomain of $\mathbb{S}^2$ supporting a hyperbolic metric.
I ...
3
votes
0answers
83 views
Coordinate-free proof of the hamiltonian character of the geodesic flow
Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the tangent and the cotangent bundles through the musical isomorphism $g^\flat:u\in TM\to g(u,\cdot)\in T^\ast M.$
It is well known ...
5
votes
1answer
105 views
Deformation of the Kähler structure on $CP^n$
Using the homogeneous coordinate on $CP^n$, we consider the open set $U_0:=\{[1, \ldots, z_n]\}$. Then the standard Kähler form of $CP^n$ can be written as
$$
...
4
votes
0answers
139 views
How does a Riemannian metric “naturally induce” distances in the Grassmannian bundle
When reading about Pesin theory I've run into the necessity of defining a metric on the Grassmannian bundle of a compact Riemannian manifold $M$. More specifically a fiber at $x \in M$ in the ...
