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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Some questions about synthetic differential geometry

I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a ...
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31 views

Distance on a 3-sphere

The arc-length $l$ between two points on on a 2-sphere of radius $R$ is given by $l=R\theta $ where $\theta$ is the subtended angle. I can rewrite this in terms of the euclidean distance $d$ between ...
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23 views

vector bundles and cocycles

I need a detailed solution to a self-study book's exercise: "Show that two vector bundles on M are isomorphic iff their cocycles relative to some open cover are equivalent" I can show it in one ...
3
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2answers
88 views

Prove the sphere is orientable

Is there an easy way to show that the sphere is orientable other then using stereograohic projection. I am preferably looking for something derived from a basic theorem in elementary geometry with ...
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vote
0answers
21 views

A curve is contained in a circle [duplicate]

I need to prove that if $\alpha: I\rightarrow \mathbb{R}^{2}$ is regular with curvature $\kappa$, then $\alpha$ is on a circle with radius $r>0$ if and only if $|\kappa(t)|=\dfrac{1}{r}$.
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1answer
35 views

Geometry, showing if F is diffeomorphism then tangent map is invertible.

Show that if f is a diffeomorphism, then Tp1(f) is invertible for all P1 member of S1. I've already proved its a linear map, I just need to show that the kernel is 0. How does one go about doing ...
3
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1answer
52 views

Integrating tensors on manifolds

When/how can you integrate tensors on manifolds and what does it mean? I imagine that line integrals of tensors make sense when you have a connection, since you can uniquely parallel transport all ...
0
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1answer
91 views

Creating topological spaces with portals

I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension. I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
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1answer
24 views

Parallel Transporting a vector

I want to parallel transport a vector $V^{\mu}$ with the initial condition $V^{\mu} = (V^{\theta},V^{\phi}) = (1,0)$ along a closed curve parameterized by $ \lambda \in [0,1]$ and determine the ...
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19 views

Make differential 1-form invariant for lift to universal cover

Suppose $ F : M \to M$ is diffeomorphism of smooth manifold $M$, and suppose $F^* \nu = \nu$ for differential 2-form $\nu$. Let $p: \tilde{M} \to M$ denote universal cover of $M$, and suppose $p^*\nu$ ...
5
votes
1answer
80 views

One-forms in differentiable manifolds and differentials in calculus

Suppose that we have this metric and want to find null paths: $$ds^2=-dt^2+dx^2$$ We can easily treat $dt$ and $dx$ "like" differentials in calculus and obtain for $ds=0$ $$dx=\pm dt \to x=\pm t$$ ...
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votes
3answers
76 views

Constructing a vector bundle using Vector bundle construction lemma

Given are: an open cover of $\{U_\alpha\}_{\alpha\in A}$ of a smooth manifold $M$. smooth maps $\tau_{\alpha\beta}\colon U_{\alpha}\cap U_{\beta}\rightarrow \text{GL}(k,\mathbb{R})$ with ...
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0answers
34 views

Counterexpample for solution for D.E. of second order

Set $M$ a $n$-manifold with $1\leq n$. Show that not every curve in $M$ is the solution for a differential equation of second order. A curve on $M$ is a differentiable fuction ...
4
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2answers
42 views

How to define continuity of functions from $R$ to $P(R^2)$?

Consider a 2-dimensional amoeba that moves in $R^2$. This amoeba can be defined as a function $f$ from a real interval to $P(R^2)$: the real interval represents the time, and $P(R^2)$ (= the subsets ...
2
votes
1answer
66 views

Physical intuition behind Green's theorem?

I get what the theorem is saying. The circulation of the curve is equal to adding up the "microscopic" curls. But why? Why does adding up the "microscopic curls" give you the circulation? Could ...
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0answers
14 views

Restrictions for Green's Theorem?

a) Why does C have to be simple? I mean the difference in circulation should be negligible if the curve only crosses itself once right? Shouldn't the condition be the curve can only cross itself a ...
2
votes
1answer
32 views

Arbitrary Smooth structure

Is it possible to give a smooth structure to any objects? Say two lines intersecting at a point. It seems there is a smooth structure though at the intersecting point it is not locally euclidean if ...
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0answers
11 views

Gauge theory H(P) and V(P)

In general you get a connection out of a one-form $\omega$ where $\omega = g^{-1}dg+g^{-1}Ag$ and $A= A^{\alpha}_{\mu}\frac{\lambda_{\alpha}}{2i}dx^{\mu}$ is given and the base ...
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25 views

Co-ordinate chart and components of a vector field.

Q) Using a coordinate chart, give a formula for the components of the vector field $[v,w]$ in terms of the components of $v$ and $w$. Where $[v,w]: f \mapsto v(wf) - w(vf)$ I don't know what the ...
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1answer
41 views

Riemann sphere, metric derivation-Completed

I have been calculated Riemann sphere, but i got stuck with calculating its metric. Consider complex plane $\mathbf{C}$ and its point $\zeta=\xi+i\eta$. And consider a point in $S^2 / (0,0,1)$ which ...
6
votes
2answers
114 views

Regularity of the heat kernel

Let $(M,g)$ be a compact Riemannian manifold. Let $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ be the heat kernel. i.e. $H\in C^0(M\times M\times\mathbb{R}_{>0})$ is the unique continuous ...
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0answers
26 views

Non-commuting flows and obtaining a new expression about the pullback of a function

If $\mathbb{X}, \mathbb{Y} \in \mathcal{X}(U)$, the set of smooth, complete vector fields on $U$ where $U$ is an open set. Let $\Phi_t,\Psi_s$ are their respective flows and let $\Gamma_{s,t}= ...
2
votes
2answers
46 views

Proving that a curve is a Geodesic in the Poincaré Half-Plane

Let $\mathbb{H}^2$ be the Poincaré Half-Plane, that is, $\mathbb{R}\times \mathbb{R}_+^*$ with the Riemannian metric $$\langle u,v \rangle_{(x,y)} = \frac{u \cdot v}{y^2}$$ I was asked (in a test) to ...
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1answer
67 views

How does the representation of co-vectors change if we change the basis of a vector space $V$?

I'm trying to understand how vectors, differential forms and multi-linear maps in general transform under change of coordinates. So I start with the simplest case of vectors. Here's my own attempt, ...
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1answer
37 views

Show that a function of four variables is independent of the first variable

If $g^{1}(x,y,r,s)$ is a function such that $g^{1}(\cdot) \in \mathbb{R}$ and suppose that $\displaystyle \frac{\partial g^1}{\partial x}+a\frac{\partial g^1}{\partial r}+b\frac{\partial g^1}{\partial ...
2
votes
1answer
39 views

Curvature of function given by points

I have function surface given by f(x,y) values (plus x and y). I know normal at each point and neighbourhood points as well. Is it possible to use this information to calculate Gaussian and Mean ...
3
votes
0answers
28 views

quaternion vector bundle and quaternion grassmannian

Let $\mathbb{H}$ be quaternion numbers. Let $G_n(\mathbb{H}^\infty)$ be the grassmannian of $n$-subspaces of $\mathbb{H}^\infty$. Then ...
1
vote
1answer
45 views

Manipulation of Tensors

I have an expression: $\eta^{\mu \nu} F_{\alpha \beta, \nu} F^{\alpha \beta}$ Where $\eta^{\mu \nu}$ is the Minkowski metric, F is an antisymmetric tensor, and the comma on the middle tensor denotes ...
0
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0answers
52 views

Lie groups, Lie algebra and left invariant vector fields

Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let $a$ and $g$ be elements of a Lie group $G$, the left ...
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0answers
17 views

Reference request: foliations

I am looking for a gentle introduction to foliations for smooth manifolds, but I have a hard time finding a textbook explaining this notion. Wikipedia's links are also to articles. Is there any ...
2
votes
1answer
22 views

Evaluating the Lie derivative of the metric

From the Wikipedia definition of the Lie derivative of a tensor along a vector field, we have, $$\mathcal{L}_X g_{\mu\nu} = X^\lambda \nabla_\lambda g_{\mu\nu} + (\nabla_\mu X^\lambda)g_{\lambda \nu} ...
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2answers
56 views

Oval/quadrupole characterization

Context: Points on a circle satisfy the equation: $$x^2+y^2=r^2$$ where $r$ is the radius. In a similar manner one can show for an ellipse: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ is satisfied($a$ ...
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2answers
39 views

Writing a two-form as a wedge product

Suppose a differential two-form $\Omega$ on $\mathbb{R}^2$ is defined by $\Omega_p(x, y)=p_2(x_1y_2-x_2y_1)$. Then using coordinates $(p_1, p_2)$ for $\mathbb{R}^2$, this reads ...
3
votes
1answer
96 views

The properness of a submersion

Let $M$ and $N$ be two differential manifolds and there is a surjective submersion $f$ from $M$ to $N$ such that $f^{-1}(p)$ is compact and connected for any $p$ on $N$. Can we conclude that $f$ is ...
6
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1answer
40 views

The existence of complete Riemannian metric

If $M$ is a differential manifold, can we necessarily find a complete Riemannian manifold on $M$? (I know we can find a Riemannian metric without completeness assumption.)
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0answers
21 views

Position Vector - Using Geometry

Given Data and Specifications in the problem $O_i,O_j$ represent two co-ordinate systems with base orthogonal frame vectors $x_j,y_j,z_j$ and another frame $x_i,y_i,z_i$ with respect to that ...
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votes
0answers
38 views

Poincare disk and mobius transformation

I have following problem Consider Poincare disk. $i.e$ $\mathbb{M}=\{ (x,y) \in \mathbb{R}^2 : x^2+y^2 <1 \}$ with metric $ g= 4\frac{dx^2 +dy^2}{(1-x^2-y^2)^2}$ Show the complex mobius ...
0
votes
2answers
63 views

computation on hyper surface $z=x^2+y^2$

I have problem with following exercise Consider the hypersurface $M$ parametrized by $z=x^2+y^2$. Endow this with the Riemannian metric induced from the $\mathbb{R}^3$. Compute the sectional ...
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0answers
28 views

homotopy via isotopy

Is it possible to realize a homotopy between two vector bundles (which are sub-bundles of the tangent bundle) over the same base $B$ as induced by an ambient isotpy? In other words, assume $X_t $ is a ...
1
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0answers
16 views

Is $z^2 = x^2 \cos y + 1$ an orientable surface?

So I can find one parametrization $\phi (u, v) = (u, v, \sqrt{u^2 \cos v + 1})$ which only does have of it. So now I need to find another parametrization which overlaps non trivally with $\phi.$ I ...
3
votes
1answer
45 views

Find $T_\mathrm{id}\left(\mathrm{Diff}(S^1)\right)$

We established on last tutorial that $T_\mathrm{id}(\mathrm{Diff}(S^1))$ are vector fields on $S^1$. I'd be grateful for any explanation (formal or intuitive) standing behind this answer.
0
votes
1answer
24 views

Defintion of Generalized Conic Bundle

Can some one help me understanding why the Definition of Generalized Conic Bundle is generalization of the Conic Bundle definition. This is the definition of a conic bundle from "Comparison theorems ...
3
votes
1answer
51 views

Question about parallel fields and geodesics

Suppose $V$ is a vector fields on a geodesic $\alpha$. Show that $V$ is parallel if, and only if $\| V\| $ is constant and the angle between $V$ and $\alpha'$ is constant. I have done the following: ...
2
votes
1answer
41 views

Poincare Hopf Theorem

I'm trying to apply the Poincare-Hopf theorem for a vector field over a closed disk. The vector fields sometimes have zeros on the boundary (if number of zeros is infinite, then it's zero over the ...
2
votes
2answers
73 views

Elementary question in differential topology

Let $p$ be a polynomial in $\mathbb{R}^n.$ Is the variety $Z(p)$ a $n-1$ dimensional manifold?another words why is that $o=(0,\cdots,0)$ a regular point for $p$ viewed as a smooth function from ...
2
votes
1answer
88 views

Expressions with the connection form in a Riemannian manifold $M^2$.

Let $M$ be a $2$-dimensional Riemannian manifold, and ${\bf x}: U \subset \Bbb R^2 \to M$ be a parametrization of $M$. Suppose that $\bf x$ is orthogonal, that is, $F = \langle {\bf x}_u,{\bf ...
3
votes
1answer
105 views

How to calculate differential forms on $S^2$ parameterized by stereographic projection?

Suppose that we have the stereographic mapping $\varphi: \mathbb{R}^2\to M$ where $M=S^2-\{(0,0,1)\}$. I've already found that the stereographic parametrization of $S^2-\{(0,0,1)\}$ is given by: ...
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0answers
42 views

A problem of Lee's 'Introduction to smooth manifolds' book

In Lee's 'Introduction to smooth manifolds', how do i prove problem $5-9$ : Suppose $\pi: M \to N$ is a smooth map such that every point of $M$ is in the image of a smooth local section of $\pi$. ...
0
votes
1answer
25 views

questions about transversal surfaces (curves) to a vector field

The following is an excerpt from Dynamical Systems by Shlomo Sternberg: By a transversal, $L$, to the vector field $V$ we mean a surface of codimension one which is nowhere tangent to $V$ . In ...
2
votes
1answer
68 views

Interesting differential equation

Given the continuous function $\mathbf{v}:I\to\mathbb{R}^2$, is it posible to solve the following differential equation: $\mathbf{v}(t)=\mathbf{u}(t)+\dfrac{\mathbf{u'}(t)}{||\mathbf{u'}(t)||}$, ...