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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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
70 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
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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
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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 ...
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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 ...
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0answers
56 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
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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
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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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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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39 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 ...
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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 ...
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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
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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.
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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
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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
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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
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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 ...
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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 ...
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1answer
109 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
44 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 $\pi$. ...
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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
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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: $\mathbf{v}(t)=\mathbf{u}(t)+\dfrac{\mathbf{u'}(t)}{||\mathbf{u'}(t)||}$, ...
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0answers
32 views

Divergence of twice contravariant tensors

In a precedent topic (which can be found here : Relationship between divergence operators defined with respect to two different volume forms. ), I asked the question of the relationship between the ...
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2answers
67 views

Meaning of Exponential map

I've been studying differential geometry using Do Carmo's book. There's the notion of exponential map, but I don't understand why it is called "exponential" map. How does it has something to do with ...
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1answer
25 views

Maximum of the length of a curve and its curvature

I'm trying to solve this problem but I can't find any way to do it. Some hints or helps will be very useful and I'll be very thankful. The problem says: Let $\alpha : (a,b) \rightarrow \mathbb{R}^2$ ...
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2answers
56 views

Relationship between divergence operators defined with respect to two different volume forms.

Let us assume that you have a volume form $\mu$ defined on a manifold $\mathcal{M}$. Then you can define the divergence operator with respect to this metric, such that the following relationship holds ...
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30 views

$C^{\infty}$-homotopy type of the Moebius band

The Moebius band $N$ has the same $C^{\infty}$-homotopy type of $S^1 \times \mathbb{R}$. What is the explicit expression of the $2$ $C^{\infty}$-homotopies involved ?
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1answer
41 views

Does this notion of the “directed area” of a closed curve in $\mathbb R^3$ have a standard name?

Given an oriented surface $\Omega$ in $\mathbb R^3$, consider the quantity $\mathbf A(\Omega)=\int_\Omega\hat n\,\mathrm dA$. We may call this the "directed area" of the surface because, when $\Omega$ ...
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0answers
34 views

Can you give a example about of curvature tensor

Can your give a Riemann manifold $(M^n,g)$,let $R(X,Y,Z,W)=g(R(Z,W)X,Y)$,and under some coframe $w^1,...w^n$, $$R=R_{ijkl}w^i \bigotimes w^j\bigotimes w^k \bigotimes w^l$$ such that,$\forall i,j ...
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2answers
90 views

Wedge product = set intersection?

In a research article [1] I found the following formulation: The wedge product may be considered as set intersection. For example, surfaces of constant $f(x,y,z)$ and surface of constant ...
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0answers
36 views

Conformal mapping

I want to show there is a coordinate system $(u_1, u_2): U \to \mathbb{R}^2$ Such that $(\mathrm{d}s)^2 = \lambda(\mathrm{d}u_1^2 +\mathrm{d}u_2^2)$ for some function $\lambda$. Professor says i ...
2
votes
1answer
91 views

Differential geometry: Conformal map

Let $f:\mathbb{R}_{>0} \times (0,2\pi) \rightarrow \mathbb{R}^3$ $$f(t,\xi) := (r(t) \cos( \xi) , r(t) \sin(\xi),z(t))$$ be a surface of revolution, where we assume that $r>0$ and ...
1
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1answer
35 views

Do closed submanifolds have nonzero curvature on a non-zero-measure set

If $M\subset \mathbb{R}^n$ is a $m$-dimensional submanifold, does the set with non-zero gaussian curvature always have $m$-dimensional Hausdorff measure greater than zero? For a manifold homeomorphic ...
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3answers
49 views

For a differentiable map $f: \mathbb{R^n}\to \mathbb{R^n}$, Show that $f^*({dy_1 \wedge\cdots \wedge dy_n})=\det(df)dx_1\wedge \cdots\wedge dx_n$

Let $f: \mathbb{R^n}\to \mathbb{R^n}$ be a differentiable map given by $f(x_1,\cdots, x_n) = (y_1,\cdots,y_n)$. Show that $f^*({dy_1 \wedge\cdots \wedge dy_n})=\det(df)dx_1\wedge \cdots\wedge dx_n$ ...
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1answer
127 views

What happens if you follow the sun?

Travelling around for quite a while and sometimes, well, just following the sun, today the question occurred to me: What happens if you really do this? So let's say some point is moving along the ...
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1answer
30 views

Equivalence of lens shaped domain and the existence of a smooth time function

A lens shape doimain is defined here as: Defn A lens-shaped domain $D\subset M$ based on $\Sigma$ is the image of a smooth map $\Phi: \Sigma\times (-1,1) \to M$ where $\Sigma\subset M$ is a compact, ...
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2answers
64 views

A compact connected solvable Lie group is a torus

I am looking for the proof of the following statement. A compact connected solvable Lie group of dimension $n\geq 1$ is a torus, i.e., it isomorphic to the product of $n$ copies of $S^1$. A ...
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2answers
136 views

why do you need to know topology to study differentional geometry

Why do I need to know topology to study differentional geometry? I just try to understand differentional geometry, but I am not sure why topology is needed for it. while I see that topology is an ...
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0answers
19 views

Christophel Symbols and planar

How can we get torsion from the christophel symbols? I want to show something is planar and am using christophel symbols, but how can I get torsion?
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0answers
34 views

Derivations on a Manifold

Let $M\subset \mathbb{R}^n$ be an $m$-dimensional manifold (in the ordinary Euclidean sense). Given a point $p\in M$ we define a derivation $D_p$ (at $p$) to be linear functional on the space of ...
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0answers
38 views

$\mathbb{C}P^1$ diffeomorphic to $S^2$

I am trying to show that the complex projective line is diffeomorphic to the 2-sphere. I'm using the $C^{\infty}$ structure on $\mathbb{C}P^1$ given by $U_1 = \{ [z_1 : 1], z_1 \in \mathbb{C} \}$, ...
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0answers
18 views

If a surface has no umbilic points how can it have planar points with a nonzero constant curvature k1

If we have a surface that has no umbilic points, but has a nonzero constant curvature $k_1$, how can there be any planar points? I am confused because for a umbilic point $k_1=k_2$ and a planar point ...
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1answer
22 views

Isomorphism of principal $G$-bundle

If $\pi: P\rightarrow M$ is a principal $G$-bundle we say $f:P\rightarrow P$ is automorphism of $P$ if $f$ is smooth, $f$ takes $\pi^{-1}(x)$ to $\pi^{-1}(x)$ and $f$ is compatible with $G$-action. ...
0
votes
1answer
16 views

How to find the reflection matrix

$V$ is an m dimensional subspace of $\mathbb{C}^n$ , n>m with an orthonormal basis {$q_1$,..,$q_m$}. How to find the reflector $P\in \mathbb{C}^{nxn}$ that reflects about $V$. $P$ must depend on ...