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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Is an isometric embedding of a disk determined by the boundary?

Suppose we cut a disk out of a flat piece of paper and then manipulate it in three dimensions (folding, bending, etc.) Can we determine where the paper is from the position of the boundary circle? ...
2
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1answer
178 views

Why is the space of all connection on a vector bundle an affine space?

I think this result is very well known, but I don't understand its proof. Let E a vector bundle over a manifold M, and $\Omega^i(E):=\Gamma(\Lambda^iT^*M\otimes E)$ the space of E-valued differential ...
0
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1answer
86 views

definition of a map from $CP^1$

I think this is a very easy question, but I've got problems understanding how the function in the second exercise of this pdf (that I found online on google and I wanted to try in order to improve my ...
3
votes
2answers
222 views

Should diffeomorphisms preserving arc length be affine?

Problem Suppose $\varphi\colon V=\mathbb R^n\to V$ be a differmorphism and $d\varphi$ is its tangent mapping. $\langle\circ,\circ\rangle$ is a nondegenerate (symmetric or symplectic) bilinear form on ...
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1answer
60 views

Using transition maps as a comparison tool between charts on a manifold.

In the wikipedia article http://en.wikipedia.org/wiki/Chart_%28topology%29#Transition_maps we read A transition map provides a way of comparing two charts of an atlas. To make this comparison, we ...
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1answer
138 views

Submanifold is complete

If $M$ is a complete manifold and $N\subset M$ is a closed, embedded submanifold with the induced Riemannian metric, show that $N$ is complete. I really don't know where to start. This is not ...
3
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1answer
87 views

D'Alembertian $\Box$

This question has to do with the D'Alembertian operator on a general manifold with a metric $g_{\mu\nu}$. I understand that the definition of the D'Alembertian is $$\Box \phi\equiv ...
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0answers
125 views

Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
2
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1answer
106 views

Metric on tangent vectors to tangent space

Let $M$ be a Riemannian manifold and $p$ be a point of $M$. Let $v$, $v'$ be tangent vectors to $M$ at $p$. Of course we have $\langle v,v'\rangle_p$ defined. Let $u$, $w$ be tangent vectors to ...
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1answer
87 views

Proof that the infinite cylinder is a regular surface.

I have to proof that the circular cylinder $M=\{(x,y,z)\in\mathbb{R}^3\mid x^2 + y^2 = r^2\}$ is a regular surface, where $r$ is a constant, $r>0$. Then I have to see also that $\mathrm x\colon ...
1
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1answer
135 views

Exterior product of n copies of 2-form

I have a problem with calculating exterior product of differential forms. Here is the problem: Let $\omega$ be a 2-form in $\mathbb{R}^{2n}$ given by $\omega=dx_{1}\wedge dx_{2}+dx_{3}\wedge ...
2
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2answers
240 views

Principal bundle automorphism generating global gauge transformations

Consider a principal $G$-bundle $P$ with connection form $\omega$. An automorphism $f$ of $P$ is by definition a (smooth) $G$-equivariant map: $f(p \cdot g) =f(p) \cdot g$ for all $p\in P$ and $g\in ...
2
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1answer
49 views

How can you express $f^*\left(\sum_{j_1,\dots,j_k} a_{j_1\dots j_k}dy^{j_1}\otimes\cdots\otimes dy^{j_k}\right)$ in terms of $dx^i$?

Suppose $f\colon M^n\to N^m$ is a map between manifolds, with $(x,U)$ and $(y,V)$ coordinates systems around $p$ and $f(p)$. How can you express $f^*\left(\sum_{j_1,\dots,j_k} a_{j_1\dots ...
2
votes
1answer
80 views

A connection over a 1-dim manifold is flat

Let $M$ be a 1-dimensional manifold and let $E$ be its vector bundle. I want to show that every connection $D$ on this vector bundle is flat. A connection $D$ is flat means that we have $$D_v D_w ...
1
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1answer
107 views

Flow of a vector field: how existence of a flow line implies existence of flow.

I am unable to see why there exists $U$ such that $\phi_t(x)$ exists for all $t\in[0,T]$. Can you please help me to understand the argument above. Thanks.
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2answers
76 views

Is not the surjective map $\pi$ associated with a vector bundle infact a bijection?

I am reading John M Lee's Riemannian Manifolds : An Introduction to Curvature, which is very well written. On page 16 : "Vector bundles are defined", quoting A (smooth) $k$-dimensional vector ...
1
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1answer
60 views

A slice orthogonal to each orbit

Assume that a compact (connected) Lie group $G$ acts on a manifold $M$. We choose a $G$-invariant Riemannian metric on $M$ and a point $p \in M$. Then using the exponential map at $p$, we can obtain a ...
3
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1answer
84 views

Differentiable parameterization of a curve $\Gamma$

If $\alpha:I\longrightarrow \Gamma$ and $\beta:J\longrightarrow \Gamma$ are two bijective differentiable regular parameterizations of the curve $\Gamma\subset\mathbb{R}^2$ (not necessarily of class ...
2
votes
2answers
295 views

Restriction of smooth functions.

Consider the following question: Suppose that $X$ is a subset of $\mathbb{R}^n$ and $Z$ is a subset of $X$. Show that the restriction to $Z$ of any smooth map on $X$ is a smooth map on $Z$. (Note: A ...
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2answers
74 views

invariant inner product on eigenspace

I have several questions about the following corollary: "Let G/H be a riemannian homogeneous space where G is a compact Lie group. Let $E_{\lambda}=\lbrace f\in C^{\infty}(G/H) : -\Delta f= \lambda ...
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2answers
136 views

Uniform convergence of constant speed $C^1$ curves with the same endpoints

This is an exercise which has been asked here also: Let $a,b\in\mathbb{R}^{2}$. Let $\{\sigma_n\}_{n=1}^\infty$ be a sequence curves $ \sigma_n:[0,1]\to{\Bbb R^2} $ such that $$\sigma_n(0)=a,\ ...
0
votes
1answer
40 views

$C^{0}$-norm of a map

I have the following question: Consider a continous map $f:M\rightarrow N$, where $M,N$ are smooth manifolds. How can one define its $C^{0}(M,N)$-norm, i.e. $||f||_{C^{0}(M,N)}$ ? Greetings, Daniel
2
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1answer
241 views

First fundamental form question.

The question I posted; $6.1.2\quad$ Show that and apply an isometry of $\Bbb R^3$ to a surface does not change its first fund. form. What is the effect of a dilation (i.e., a map $\Bbb R^3\to ...
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2answers
703 views

Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
0
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1answer
63 views

Why do the coordinate functions $x^1+W,\dots,x^n+W$ form a basis of $\mathscr{F}_p/W$?

I'm a bit stuck in the following situation. Suppose $\mathscr{F}_p$ is the set of smooth functions $f\colon M\to R$ with $f(p)=0$. Also, let $W$ be the subspace generated by $fg$ for ...
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2answers
746 views

Shell method for calculating volume of solid of revolution - general

Let us have an injective continuous function $f : [a,b] \to [0,c]$ (such that $f(a)=0$ and $f(b)=c$). I want to calculate the volume of solid revolution of $f$ around the $y$ axis. The first method ...
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235 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
2
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1answer
55 views

Why is it true that ${(\nabla Y)'^i}_j = {Y'^i}_{,j} + {\Gamma'^i}_{jk}Y'^k $?

I do not understand why this equation transforms as it does : ${(\nabla Y)'^i}_j = {Y'^i}_{,j} + {\Gamma'^i}_{jk}Y'^k $ Could someone give me a detailed explanation of why this is true please? I ...
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0answers
55 views

Doubt on the definition of topological manifold

I need some clarifications on topological manifolds. My professor has defined them as second countable, connected, Hausdorff topological spaces which are locally homeomorphic to $\mathbb{R}^n_+$, ...
3
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0answers
147 views

Zariski cotangent space, as defined in Arapura's “Algebraic Geometry over the Complex Numbers”

In "Algebraic Geometry over the Complex Numbers", Arapura gives the following definition: Definition 2.5.8. When $(R, m, k)$ is a local ring satisfying the tangent space conditions, we define its ...
2
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1answer
66 views

Notation in Bleecker's Gauge Theory and Variational Principles

In the proof of the theorem that there is a unique linear isomorphism $\star:\bigwedge^k(E)\to\bigwedge^{n-k}$ on p.4 in Bleecker's Gauge Theory and Variational Principles he says For ...
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1answer
48 views

Why does an operator on smooth functions vanishing at $p$ have a unique extension to a derivation?

On page 78 of Spivak's Differential Geometry, he mentions that a linear operator $\ell$ on all $C^\infty$ functions is a derivation at $p$ when satisfying $$ \ell(fg)=f(p)\ell(g)+g(p)\ell(f). $$ ...
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4answers
778 views

How to convince a high school student that differentials don't work like fractions in general?

It all started when I tried to convince a 10th grader that if $f$ is a function defined on $\mathbb{R}^n$ the differential is defined by: $\large \displaystyle df = ...
6
votes
3answers
917 views

Differential Forms and Vector Fields correspondence

Barrett O'Neill's Differential Geometry book says that Classical vector analysis avoids the use of differential forms on $\mathbb{R}^3$ by converting 1-forms and 2-forms into vector fields via the ...
2
votes
1answer
106 views

Proving that two sets are diffeomorphic

I have the following two sets $\mathcal{S}= \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$ and $\mathcal{S}' = \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid ...
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1answer
136 views

Derivation for affine connection formulas on differentiable manifolds (General tensors)

Let $p\in U\subseteq M$ be a point in some neighborhood of a finite-dimensional differentiable manifold, $\{x^i\}$ a set of local coordinates with respect to $U$, and ...
1
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1answer
82 views

Metric tensors. Have I got the correct understanding?

My course is covering metric tensors in a slap-dash way, so I want to ensure I have understood correctly how they are described. I hope you can confirm this! So, I believe that a metric tensor is a ...
0
votes
1answer
62 views

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature.

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature. I dont have enough idea. Please explain the question clearly. ...
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1answer
100 views

A curve is regular if and only if its veloctiy vector is not equal to 0

I am starting to learn about Differential Geometry. I am using "Elements of Differential Geometry" by Richard S. Millman and George D. Parker (I have very little background knowledge on the subject). ...
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0answers
132 views

How is Euler-Lagrange equation used to find optimal solutions in minimizing a function?

How is the Euler-Lagrange equation: $$ L_x(t,q(t),q'(t))-\dfrac{d}{dt}L_v(t,q(t),q'(t))=0 $$ used mathematically in finding the optimal solutions of minimising a function? Can someone give me an ...
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0answers
174 views

The Fubini-Study metric and its associated form.

Assume $ds^2$ is Fubini-Study metric, and $\omega = -\frac{1}{2}Im \{ds^2\}$ is its associated form. Let $V_n = \int_{\mathbb{C}P^n} \omega^n / n!$ the volume of $\mathbb{C}P^n$. Let $V \subset ...
9
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1answer
214 views

The Curve in $R^n$

Let $r:(a,b)\rightarrow{R^n}$ with $|r^{'}|=1$ is a natural parameter curve in $R^n$. If $e_1(s)=r'(s),e_2(s),...,e_n(s)$ form an orthonormal frame, then we have Frenet formulae: ...
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votes
3answers
99 views

Geometric interpretation of deriavative of a function of more than one variable

A function $f$ is defined on an open set $D$ of $\mathbb R^{2}$ is called a differentiable at a point $x\in D$ if there is a vector $m \in \mathbb R^{2} $ such that $$\lim_{h\to 0} ...
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0answers
47 views

geometry on a torus ( or other regular non-constant Gaussian curvature planes)

Can somebody point me to a publication on geometry on a torus (or an other regular non-constant Gaussian curvature plane, i just think a torus is the simplest form of it) I want to learn more about ...
4
votes
2answers
149 views

when can you estimate curvature from finite information about two geodesics?

Let $c_v, c_w$ be two geodesics starting at a point $p\in M$, where M is a nonpositively curved, complete, smooth Riemannian manifold. Say $c_v(\varepsilon) = \exp_p(\varepsilon v)$ and ...
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1answer
847 views

Riemann tensor in terms of the metric tensor

The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used. Most tensor notation based texts give the Riemann tensor in terms of the Christoffel symbols, which are ...
2
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1answer
78 views

Relation between $L^1(\partial\Omega)$ and the surface integral on $C^1$ domains

Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain. It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite ...
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1answer
51 views

Doubt on differential

I've got a doubt on a definition on my notes: "The differential is the operator $d: C^\infty(U)\rightarrow \Omega^1(U)$ defind by the formula $df(v)=v(f) \quad (1)$". Here $U$ is an open set of ...
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0answers
310 views

The curvature and torsion of the tangent indicatrix

Let $\alpha$ be a unit speed curve. Its tangent indicatrix $\sigma$ is defined by $\sigma(t)=T(t)$. Find torsion and curvature of $\sigma$ with respect to the torsion and curvature of $\alpha$. ...
4
votes
2answers
169 views

Why do differential geometry textbooks bother with equivalence classes of smooth structures?

In contemporary textbooks on differential geometry, the definition of smooth manifolds is given in a (IMHO) awkwardly obfuscated way, by saying that a smooth manifold is a topological space endowed ...