1
vote
0answers
9 views

Fundamental solution of Laplacian on manifold

I'm looking for a reference for the result that there exists a fundamental solution for the Laplacian on a flat torus $$\Delta \Gamma(x-y) = \delta(x-y), \quad x,y \in \mathbb T^2.$$ and that, ...
1
vote
1answer
41 views

Differential-geometry textbook with solved problems

I'm looking for a textbook in differential geometry which inside has exercises with (at least) final answers. Since it's my first course in differential geometry it doesn't have to cover material (we ...
6
votes
1answer
142 views

Searching for a thesis.

In several articles about curves, etc, in Minkowski spaces $\Bbb L^3$ and $\Bbb L^4$, there is Walrave, J., Curves and surfaces in Minkowski space. Doctoral Thesis, K.U. Leuven, Fac. of ...
1
vote
1answer
30 views

Is there a Poincare lemma for codifferential?

Is every co-closed form also locally co-exact? That is for each $k$-form $\omega$ such that $\delta \omega = 0$ there exists $(k-1)$-form $\eta$ for which locally $\omega = \delta \eta$. My current ...
3
votes
1answer
61 views

Special reference for differential geometry

I am not entirely sure how to formulate the question, but here it is. I am looking to start a self study on general relativity, and of course I need a good grasp on semi-riemannian geometry (I am ...
7
votes
1answer
103 views

Is there a codifferential for a covariant exterior derivative?

For forms on a Riemannian $n$-manifold $(M,g)$ there is a notion of a codifferential $\delta$, which is adjoint to the exterior derivative: $$\int \langle d \alpha, \beta \rangle \operatorname{vol} = ...
2
votes
1answer
35 views

Identities for differential forms and vectorfields (reference request)

Recently I found the slides of a talk of J. E. Marsden, (Differential Forms and Stokes' Theorem). These slides introduce the required objects and summarize the basics of the corresponding theory. In ...
0
votes
0answers
29 views

Relationship between differentiation and integration of vector fields?

Let $V\in\Gamma(T\mathbb{R}^n)$ be a vector field and $\gamma:[a,b]\to \mathbb{R}^n$ a curve. Let $\nabla$ be the Euclidean connection, i.e. $\nabla_XY=XY^k\frac{\partial}{\partial x^k}$. We have a ...
0
votes
1answer
51 views

What is a squashed 3-sphere?

I have found the term "squashed 3-sphere" used in the literature but could not locate a precise definition of it. I suppose it is topologically a 3-sphere with a metric different from the round one. ...
1
vote
1answer
27 views

A Couple Formulas in Besse's “Einstein Manifolds”

In Besse's "Einstein Manifolds," Chapter 6D, there are 2 formulas which I am interested in, which apply to compact Riemannian $4$-manifolds: ...
2
votes
0answers
58 views

Sine-Gordon Equation application

Is it true that Sine-Gordon is satisfied for geodesics on the central Pseudosphere ( rotated surface of Tractrix)? If so, please cite text-book or article references. $$ \alpha''(s)=sin ( \alpha) ...
4
votes
2answers
105 views

How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
2
votes
1answer
45 views

1D manifold is diffeomorphic to $\mathbb R$ or to $S^1$

In his ODE classic V.I. Arnold considers easy to see (легко видеть) that every one-dimensional (connected and without boundary) differentiable manifold is either diffeomorphic to $\mathbb R$ (if it is ...
4
votes
0answers
49 views

The Kähler form and the anticanonical line bundle

Let $M$ be a Kähler manifold. We say that $M$ is Fano if the anticanonical line bundle $K_M^*$ of $M$ is ample (or positive). On the other hand, I sometimes see the following definition (or ...
2
votes
1answer
55 views

Unit circle can't be covered by one chart

I am hoping that someone can give me a proof showing why the unit circle cannot be covered by one coordinate chart, or a reference where I can find a proof.
1
vote
0answers
70 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
5
votes
0answers
52 views

Integration of bundle-valued differential forms

The literature, at least textbooks, seems to be very scarce on the topic of integrating bundle-valued differential forms. So I wonder where can I read on the topic? I want to see usual theorems, like ...
1
vote
3answers
80 views

May directed graph be embedded into manifold?

May directed graph be embedded into manifold?How ?and what is the condition?
2
votes
0answers
35 views

Reference for Envelope, Evolute and involute

I have to give a lecture on Envelopes, Evolute and Involute to I year undergraduate students. Please suggest me some books which explain these concepts with examples geometrically. Already I have seen ...
3
votes
2answers
150 views

Book for Undergrad Differential Geometry

I am soon going to start learning differential geometry on my own (I'm trying to learn the math behind General Relativity before I take it next year). I got the sense that a good, standard 1st book ...
3
votes
0answers
81 views

Complex structures on Riemann surfaces

Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq H^0(M;K^2)$. Considering $\alpha$ as a map $T^{0,1} M \to T^{1,0} M$, the bundle $$ \{v + \alpha(v) \mid v \in T^{0,1} M\} ...
2
votes
1answer
34 views

Ergodic properties of orthogonal group $O(n)$

The orthogonal group $O(n)$ is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing ...
0
votes
1answer
46 views

Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties $g$ is differentiable and injective $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb ...
5
votes
1answer
235 views

Russian Texts on Geometry

I recently saw a question today pertaining to Russian mathematics and I have a similar question but of a slightly different flavor. I've always heard that the Soviet Union had a history of producing ...
1
vote
0answers
61 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 ...
2
votes
0answers
65 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
0
votes
0answers
30 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
1
vote
0answers
18 views

Citing a result on obstruction to Lagrangian Embedding

Let $L$ be a closed orientable Lagrangian embedding in $\mathbb C^n$. Then $\chi(L) = 0$, where $\chi(L)$ denotes the Euler characteristic of $L$. This fact is more or less stated in section 3.2 of ...
5
votes
0answers
59 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
57 views

Book on Riemann geometry. [duplicate]

I am looking for good books on manifolds ,Riemann geometry .can you help me ?(I am a undergraduate level student)
1
vote
1answer
76 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 ...
3
votes
0answers
99 views

Developing intuition in algebraic geometry through differential geometry?

I'm interested in algebraic geometry (I am working through Ravi Vakil's notes and also have worked with curves and general varieties in the past), and have seen some basic definitions from ...
38
votes
5answers
1k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
0
votes
1answer
53 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. ...
2
votes
1answer
126 views

What should a student (with algebraic-geometry minded) study in differential geometry?

One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a ...
1
vote
2answers
63 views

Need help on books on diff. equations/geometry and theoretical computer science

I am looking for recommendation of 3 different books on the following topics: 1.Differential Equations -Ordinary diff. equations -Vector field, transport equations -Equation of wave and heat -Use ...
0
votes
0answers
28 views

References for Conjugate Points in Differential Geometry

I will have to give some lectures about conjugate points and I need some nice references about it, can anyone recommend me some? I already know manfredo's differential geometry of curves and surfaces ...
5
votes
3answers
256 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. ...
2
votes
0answers
61 views

A few identities on differential forms

I just started studying differentiable manifolds, and I've encountered a few allegedly simple questions that aren't really simple to me. If possible, I'd be very happy to see full proofs (without ...
6
votes
1answer
156 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}+ ...
2
votes
0answers
52 views

Application of Kodaira Embedding Theorem

I am going to give a talk on Kahler manifold. In particular, I will outline a proof of the Kadaira Embedding theorem. I also wish to give some applications of the theorem. One of the application ...
3
votes
1answer
70 views

A literature reference for Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth Riemannian manifolds

Anyone know a respectable reliable reference for the definition of Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth compact Riemannian manifolds. It suffices for m natural and $p\geq 1$
4
votes
4answers
99 views

Is there a shorter path to these results?

I'm a student of Physics, however I usually study mathematics on texts aimed at mathematicians to gain a deeper understanding. Currently I'm studying differential geometry on Spivak's book and one of ...
1
vote
2answers
147 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 ...
1
vote
1answer
46 views

Parabolic Cusp of an Action on the Upper Half Plane

This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta ...
0
votes
1answer
105 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
1
vote
0answers
37 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 ...
3
votes
2answers
144 views

Geodesic equations and christoffel symbols

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)
4
votes
1answer
81 views

Calculus on Manifolds - operational point of view

I'm a student of Physics and I've been studying manifolds and calculus on such objects for a time. Usually when we deal with vector calculus there are books that bring one operational point of view. ...
6
votes
2answers
157 views

Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such ...