10
votes
2answers
237 views

What meaning did Riemann assign to $dx$?

Detlef Laugwitz wrote a monumental biography of Riemann. The book was translated into English by Shenitzer. Laugwitz discusses Riemann's fundamental essay Uber die Hypothesen, welche der Geometrie ...
1
vote
0answers
49 views

Reference for construction on Riemannian Manifolds

I would like to know a book or an article where the connected sum of Riemannian manifolds is explained with some detail. I roughly know how to do the construction but I want to be able to check a more ...
4
votes
0answers
51 views

Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
1
vote
1answer
71 views

Reference about Gauss-Bonnet-Chern theorem.

I would like to get some references which explains Gauss-Bonnet-Chern theorem and its original proof by Chern. I tried to read his paper published in 1944 "A Simple Intrinsic Proof of the Gauss-Bonnet ...
0
votes
1answer
47 views

Length Minimizing Properties of Geodesics on Surfaces?

Can anyone recommend me some nice references about lengh minimizing properties of geodesics? I'm looking for a treatment in the case of surfaces, but more general viewpoints will also be welcome. ...
5
votes
1answer
86 views

Explicit description of flat connections under pullback on principal bundles over Riemann surfaces

I'm trying to find a proof/reference for a statement that I've seen quoted in some way or the other, but without reference. The setting: let $P\longrightarrow M$ be a flat principal $G$-bundle over ...
1
vote
1answer
60 views

Reference - Riemannian Orbifolds

I am looking for papers or textbooks talking about the various analog theorems of Riemannian Geometry of Manifolds to Riemannian Orbifolds like Toponogovs Theorem, Bonnet-Myers, Gauss Bonnet etc. So ...
5
votes
3answers
202 views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
1
vote
1answer
53 views

Request for online reference to Hamilton's “The Ricci Flow on Surfaces”

Does anyone know of an online source for Richard Hamilton's paper "The Ricci Flow on Surfaces?" I've searched Google for it and it doesn't seem to give any results.
6
votes
1answer
137 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
1
vote
2answers
136 views

Why do people stick with Riemann-Integration when dealing with differential geometry?

I asked a question yesterday that is, "Is there an introductory differential geometry text using Lebesgue integration?" Then, i got an answer that "since we are dealing with differential geometry we ...
0
votes
0answers
52 views

is there an introductory differential geometry text using Lebesgue integration?

Is there an introductory differential geometry text using Lebesgue integration? Every differential geometry text I saw introduces the theory using Riemann integration. (Even Spivak) Would someone ...
1
vote
0answers
36 views

Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
0
votes
2answers
47 views

Riemann Sphere/Surfaces Pre-Requisites

I have recently developed a large interest in everything to do with Riemann Sphere/Surfaces. I wish to understand the topic quite well but I know that I will need to read a good number of books on ...
0
votes
0answers
21 views

Concerning geodesic representatives of singular homology classes on compact Riemannian manifolds

I am currently reading a book on geodesic flows where I found the following (unproved) claim: "If $M$ is a compact Riemannian manifold then any nontrivial $\alpha \in H_1(M, \mathbb{Z})$ contains a ...
1
vote
1answer
30 views

Any books on isospectral manifolds?

I was searching stuff related to M.Kac's famous question "Can one hear the shape of the drum ?" I further found results due to Gordon, Webb and Wolpert in the 2D case using Sunada method. Are there ...
2
votes
3answers
250 views

Resources for properly developing a modern understanding tensors

I am currently learning about tensors as they come up in the mathematics behind continuum mechanics. I was fairly disappointed with my initial foray into tensors, as presented in the book Classical ...
1
vote
0answers
79 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
1
vote
1answer
50 views

generalized warped product

Let $(M_1,g_1)$, $(M_2,g_2)$ denote two Riemannian manifolds, let $(I,dt^2)$ be the unit interval with its standard metric. I would like to study the manifold $(M,g)$ where $M = I \times (M_1 \times ...
3
votes
0answers
44 views

Reference on Grassmanian

Can anyone suggest a reference on the Grassmanian which describe the Riemannian structure of the Grassmanian $Gr(k, n)$? Specifically, I want to know about the geodesics, convex neigboorhood, geodesic ...
1
vote
0answers
45 views

Suggestion for a good book that explain Cartan's Moving Frame and Riemannian Geometry

I'm studying Riemannian Geometry, and I'm having a lot of trouble with the book Riemannian Geometry and Differential Dorms both from do Carmo.And I would like a book with examples, calculations, if ...
4
votes
1answer
117 views

What groups can be realized as the isometry group of the two-sphere?

Regarding $S^2 \subseteq \mathbb{E}^3$ as a Riemannian manifold with the inherited metric from Euclidean three-space, then it is well known that the isometry group is $O(3)$. What I am curious about, ...
1
vote
0answers
130 views

Ambrose Singer Theorem

I wish to learn about holonomy groups of Riemannian manifolds and the Ambrose- Singer theorem. Please advise some references other than the original paper of Ambrose and Singer.
6
votes
3answers
428 views

Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
3
votes
0answers
114 views

Every holomorphic map between Kähler manifolds is harmonic

I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section: Every holomorphic map between Kähler manifolds is harmonic. I am not that familiar ...
3
votes
1answer
134 views

Need help finding a good book on Riemann Geometry

I want to learn more about calculus on manifolds and Riemann Geometry. I have been reading the book Geometry, Topology and Physics by Nakahara. But I find that it is difficult to read due to the lack ...
8
votes
1answer
200 views

Creating geodesics on manifolds

Suppose I have two points on a Riemannian manifold $M$, called $p_0$ and $p_1$. I have a family of curves $\gamma:[0,\infty)\times[0,L]\to M$ such that $\gamma(t,0) = p_0$ and $\gamma(t,L) = p_1$. ...
1
vote
1answer
57 views

Boundary Normal Coordinates

Let $ (M,g) $ be a 2D Riemannian manifold with boundary. The boundary normal coordinates $\psi $ are constructed roughly as follows: in a sufficiently small neighborhood $ U $ of $ \partial M $, for ...
3
votes
0answers
68 views

Distance function and Green's function

Let $ (M,g) $ be a two dimensional Riemmanian manifold, with a smooth distance function $ d(x,y) $ for all $ x, y$ in $ M$. The logarithm of the distance function then satisfies $ \Delta \left( ...
5
votes
3answers
234 views

Reference request for some topics in Differential Geometry like connections, metrics, curvature etc.

I wish to study some topics in Differential Geometry like Vector bundles, Metrics, Connections, Curvature,Geodesics, Holonomy, Characteristic classes etc. I am considering reading this book ' ...
1
vote
1answer
37 views

Reference request for studying on space forms

I would like to study on Space form, But I dont know what book or notes are suitable for beginning basically. Can someone help me? Thanks.
5
votes
0answers
87 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
105 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
3answers
356 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
135 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 ...
4
votes
2answers
634 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
103 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
280 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: ...
3
votes
1answer
353 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 ...
2
votes
1answer
266 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
261 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
581 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
136 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
234 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 ...
7
votes
1answer
166 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
549 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
183 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 ...
2
votes
1answer
305 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
307 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
109 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 ...