Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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743 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) ...
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1answer
326 views

interior product - proof of the basic fomula

How would you prove the interior product formula? Namely for $\omega \in \Omega^k (X), \mu\in \Omega^l(X)$, where $X$ is smooth manifold with vector field $v$ we have $$i(v)(\omega ...
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1answer
84 views

How do I find the number of vertices on a planar diagram of a surface?

Cheers, I have a question which I just do not seem to see the answer for: I am proving the classification theorem for compact surfaces and use planar diagrams as representation of the surfaces. I ...
1
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1answer
161 views

The number of geodesics of a complete Riemann manifold with non-positive sectional curvature

There is a theorem of Cartan which states that if $M$ is a simply connected, complete Riemann manifold, and that the sectional curvature is everywhere $\leq 0$, then any two points of M are joined by ...
2
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0answers
61 views

Unit partition to produce smooth function from continuous ones

Given a positive continuous function (except on closed set, where is zero ) on a smooth manifold how to find a smooth function under the same conditions being less (or equal) than this one ...
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votes
3answers
410 views

Introductory Treatment of Differential Geometry

I'm going to be taking a graduate course in differential geometry, this coming fall, but I am not prepared for it. Can anyone recommend a good introductory treatment of the background materials? The ...
2
votes
3answers
311 views

Complete developable surface in $\mathbb{R}^3$ is ruled

Let $X \subset \mathbb{R}^3$ be a complete smooth surface which is developable in the sense that its Gaussian curvature is identically zero. Wikipedia claims that such a surface is necessarily ruled, ...
2
votes
2answers
438 views

Openness of $\varphi(U_Q \cap U_{Q'})$ in the definition of Grassmannian Manifolds (Lee: Introduction to Smooth Manifolds)

I am reading Lee's Introduction to Smooth Manifolds and I have some problems with definition of Grassmannian manifold given in Example 1.24, p.22. I'll write the details below. My question is: Why ...
7
votes
1answer
566 views

how to understand the tensor product canonical line bundle $\otimes$ dual bundle

Suppose we have a Riemann surface $M$ together with a holomorphic vector bundle $E \to M$ of rank n. let $K$ denote the canonical line bundle and let $E^*$ denote the dual bundle I am trying to ...
4
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0answers
221 views

A curve in a submanifold with a tangent vector not necessarily in the submanifold's tangent space

I came across this exercise in Warner's Foundations of Differential Manifolds and Lie Groups, Let $N \in M$ be a submanifold. Let $\gamma \colon (a,b) \to M$ be a $C^{\infty}$ curve such that ...
2
votes
2answers
597 views

minimal surface of revolution when endpoints on x-axis?

What is the formula for the planar curve through $(\pm a,0)$ of fixed length $l$ which has minimal-area surface of revolution when rotated about the x-axis? I get the area of the surface to be ...
5
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0answers
355 views

Definition of Coadjoint representation for Lie algebras

I have trouble understanding the definition of the coadjoint representation of a Lie algebra. Typically you first define a natural pairing between the Lie algebra and Lie coalgebra: \begin{equation} ...
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2answers
478 views

Index notation and differentiation

Let $x_i$ such that $i=1,2,\ldots,n$, and $\vec{x}=(x_1,\ldots,x_n)$ Define $$A:= M_{ij}(\vec{x})\dot{x}^i\dot{x}^j$$ where Einstein summation applies. Also, $M$ is symmetric and invertible -- a ...
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1answer
448 views

How to show that a map is an isometry

I'm having a difficulty understanding how to go on proving a certain map is an isometry. It should be really basic and simple, but for some reason I can't understand how to do this.. The situation ...
5
votes
3answers
662 views

Physical interpretation of the Lie Bracket

I've come accross this physical interpretation for $ [X,Y] $ which I don't understand : Follow $X$ for some time $\epsilon$; Follow $Y$ for $\epsilon$; Follow -X for $\epsilon$; Follow -Y for ...
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1answer
233 views

a question about definition of regular surface

While I am reading Do Carmo's differential geometry,I have several questions about the definition of regular surface. From condition 2,the author said : "...... $x^{-1}:V \cap S \rightarrow U$ ...
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0answers
48 views

why do i need convexity of the set of positive definite hermitian matrices for hermitian structure on vector bundle?

Consider a Riemann surface $\Sigma$ and a holomorphic vector bundle $\mathcal{E} \to \Sigma$ of rank $N$. I just came across the remark that in order to endow $\mathcal{E}$ with a Hermitian metric it ...
7
votes
1answer
244 views

support of a differential form on manifold

In the book "Differential forms in Algebraic Topology" by Bott and Tu, the support of a differential form $\omega$ on a manifold $M$ is defined to be "the smallest closed set $Z$ so that $\omega$ ...
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1answer
229 views

Partition of Unity question

I am starting to read the book "Differential Forms in Algebraic Topology" by Bott and Tu. In the proof of the exactness of the Mayer - Vietoris sequence (Proposition 2.3, page 22 - 23) a partition ...
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1answer
158 views

Question about an isometric immersion

This is the question: Let $M,N$ be Riemannian manifolds, such that the inclusion $i:M \to N$ is a isometric immersion. Give a example where the inequality $d_M > d_N$ may occur. I thought ...
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votes
2answers
127 views

Parametrise curve by angle and convex curves

Can one parametrise any closed curve by the angle its tangent makes to the $x$-axis? I seem to remember that this is only possible for convex curves. Could anyone tell me why, please? Also is ...
7
votes
2answers
415 views

Definition of the Lie coalgebra

I don't understand how the Lie coalgebra is defined. The literature is never really explicit in how it is constructed. So I was wondering if anybody could supply me with a simple example of how the ...
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2answers
213 views

Immersed curve in $\mathbb{R}^2$ and regular curve

I read that a curve $\gamma:S^1 \to \mathbb{R}^2$ is an immersion iff it's regular. So it's an immersion iff $\gamma'(t) \neq 0$ for all $t \in S^1$. An immersed curve is one whose derivative is an ...
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2answers
366 views

To show $X$ is a complete vector field on $M$

Well, I have solved myself the problem : every smooth vector field on a compact manifold is complete. Now I have got this problem which I am not able to progress: let $X$ is a vector field on $M$, ...
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0answers
88 views

Manifold contains a totally geodesic closed hypersurface

Let $(M^n,g)$ be a closed simply-connected positively curved manifold. Show that if $M$ contains a totally geodesic closed hypersurface (i.e., the second fndamental form or shape operator is zero), ...
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0answers
89 views

uniqueness of asymptote in manifold

Question 1 Let $M$ be a complete, noncompact Riemannian manifold, a ray $\gamma:[0,\infty) \rightarrow M$ starting from $p$, and a point $x \in M$ such that the asymptote $\widetilde{\gamma}$ ...
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votes
1answer
157 views

Possibilities of an action of $S^1$ on a disk.

I'm dealing with actions of the circle over differentiable manifolds. In the book I'm reading, they use the fact that an action of $S^1$ over a disk has to be equivalent (there has to exist an ...
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1answer
84 views

Sard's Theorem For Constant functions

It states: Let $g:A \to R^n$ be continuously differentiable, where $A \subset R^n$ is open, and let $B=${${x \in A: \det g'(x)=0}$}. Thne $g(B)$ has measure $0$. Okay.... obviously this theorem is ...
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1answer
75 views

Changing domain of PDE (should be easy)

Suppose I have a PDE $$u_t(x,t) + f(x,t)\cdot \nabla u(x,t) = 0 \quad \text{on $\gamma(t)$}$$ where $\gamma(t)$ is a curve for each fixed t in $\mathbb{R}^2$ and $f$ is given. We have that ...
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1answer
465 views

smooth maps between smooth manifolds - Jacobian coordinate independence

I have question about comment in Lee's Introduction to Smooth Manifolds - page 51. Given smooth map $F:M\to N$ between smooth manifolds $M$ and $N$ we say that the total derivative of $F$ at $p\in ...
6
votes
2answers
647 views

Poincaré Lemma Contractible Hypothesis

Poincaré's Lemma is often stated as saying that a closed differential form on a star-shaped domain is exact. More generally, it is true that a closed differential form on a contractible domain is ...
4
votes
1answer
993 views

Problem book on differential forms wanted

I want to get used to differential forms. Thus I would like to solve a bunch of problems, especially on integration of differential forms. So I need a collection of problems with answers/solutions, ...
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0answers
131 views

A series of Lemmas about $C^{\infty}$ vector fields

Warner page-37: Theorem: Let $X$ be a $C^{\infty}$ vector field on a differentiable manifold $M$ For each $m\in M$ there exist $a(m)$ and $b(m)$ in extended real line, and smooth curve ...
2
votes
1answer
74 views

Differentiable map conserving geodetic lines which is no isometry

I am looking for a differentiable map $f: S^n\rightarrow S^n$, which conserves the geodetic lines of the standard metric on $S^n$, but is no isometry. The geodetic lines on $S^n$ should be the great ...
2
votes
2answers
297 views

Generalized Laplacian operator?

Suppose a surface $S$ is endowed with a metric given by the matrix $$M=\begin{pmatrix} E&F\\F&G\end{pmatrix}$$ And $f,g$ are scalar functions defined on the surface. What then is the ...
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votes
1answer
136 views

Left and Right Vector bundles

I am reading a paper that starts talking about 'left vector bundles' and I'm having trouble figuring out what they mean. The specific setup is as follows: A quarternionic line bundle $L$ over ...
2
votes
1answer
157 views

Chain rule and gradient

Let $\Gamma \subset \mathbb{R}^2$ be a curve. Define for a smooth function $f$, $$\nabla_\Gamma f = \nabla f - (\nabla f \cdot N)N$$ where $N$ is the unit normal. Let $X:S \to \Gamma$ be a smooth ...
0
votes
1answer
133 views

A $C^\infty$ map $M^n \to \mathbb R^n$ must have singular points if $M$ is compact

Can anyone give me an hint: If $M$ is a compact manifold of dimension $n$ and $f:M\rightarrow \mathbb{R}^n$ is $C^{\infty}$, then $f$ can not be everywhere nonsingular. Suppose $f$ is everywhere ...
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0answers
95 views

The Implicit Function Theorem and open sets with regular boundary

Let $\rho :\mathbb{R}^N\longrightarrow\mathbb{R}$ be a continuously differentiable function such that $\rho (x) = 0 \Rightarrow d\rho (x) \not = 0$ for all $x\in\mathbb{R}^N$. Suppose $\Omega ...
0
votes
1answer
146 views

doubt on $C^{\infty}$ vector field

Warner page-37: Theorem: Let $X$ be a $C^{\infty}$ vector field on a differentiable manifold $M$ For each $m\in M$ there exist $a(m)$ and $b(m)$ in extended real line, and smooth curve ...
1
vote
1answer
313 views

How long will it take me learn and understand differential (Riemannian) geometry for PDEs? [closed]

I want to learn DG and RG so I can use them in PDEs. Atm I have no knowledge of either DG or RG (and not that much of PDEs either..) but I have a couple of books (John M Lee and Loring). If I spend ...
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vote
1answer
59 views

non-linear partial differential operators

I am looking for some literature on non-linear partial differential operators used in geometry or analysis. Can you give me some reference. Thanks in advance. eric
4
votes
2answers
302 views

Self-Linking Number on 3-Manifolds

We can assign a framing to a knot $K$ (in some nice enough space $M$) in order to calculate the self-linking number $lk(K,K)$. But of course it is not necessarily canonical, as added twists in your ...
0
votes
1answer
70 views

Non connected surfaces and Gauss Bonnet Theorem

In general, I have seen that a consequence of the Gauss-Bonnet Theorem is the following: Theorem. If S is a CONNECTED smooth compact oriented surface in $R^3$, then S is diffeomorphic to a $g$-tori ...
2
votes
1answer
119 views

question about integral curve on a Manifold

well In warner book, page 36. A curve $\gamma:(a,b)\rightarrow M$ is integral curve iff $$d\gamma(\frac{d}{dr}|_t)=X(\gamma(t))$$ Could anyone explain me about the left side in a detail and breaking ...
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0answers
112 views

Dimension of diffeomorphism groups preserving some $2$-tensor.

For a finite-dimensional smooth manifold $M$, let $\mathrm{Diff}(M)$ be its diffeomorphism group. Suppose we are given a $2$-tensor $\mathcal{K}$ on $M$, and let $$\mathrm{Diff}_{~\mathcal{K}}(M) = ...
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1answer
272 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 ...
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1answer
237 views

Gradient in curvilinear coordinates

The gradient is usually written as the product of the unit vectors times the derivative with respect to that coordinate. In Einstein summation convention: $\hat e_i \partial_i$ I've seen it written ...
3
votes
0answers
250 views

Vector bundle with Moebius strip as base space

This question has a motivation in physics, thus its formulation may not be entirely rigorous. Let $f$ be a function that takes values on a Moebius strip of fixed length $L$ and maps them to ...
5
votes
1answer
265 views

Why does the Gauss-Bonnet theorem apply only to even number of dimensons?

One can use the Gauss-Bonnet theorem in 2D or 4D to deduce topological characteristics of a manifold by doing integrals over the curvature at each point. First, why isn't there an equivalent theorem ...