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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Problem finding regular values of map

I found an exercise defining $f:S^3\to\mathbb{CP}^1$ by $f(x,y,z,t)=[x+iy:z+it]$ and asking to prove it was smooth and find its regular values. Proving it was smooth was simple enough. Then I tried ...
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63 views

Curvature in $\mathbb{R}^2$

Let $f(t) = (x(t),y(t))$, not necessarily parametrized by arclength. We define the unit tangent vector, $T(t) = (1/|f'(t)|)(x',y')$. Also the normal vector, $N(t) = (1/|f'(t)|)(-y',x')$, which is ...
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fiber bundle in topological category and smooth category.

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by ...
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16 views

Space curve torsion

Hello I am looking for anyone to maybe look over my ideas and see if they think it is correct. Say I am looking for the torsion $\tau$ of a space curve given by $r(t)=(cos(3t),sin(3t),4t)$ I know if ...
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526 views

Smooth maps (between manifolds) are continuous (comment in Barrett O'Neill's textbook)

(Needless to say, I'm a total newbie in differential geometry so I apologize if this seems rather too obvious to many of you). As a comment on his definition of smooth mapping, Barrett O'Neill in his ...
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136 views

Is there geometric interpretation to Skew symmetric coefficient matrix,

We know that the Frenet-Serret equation implies that the coefficient matrix of $\dot t,\dot n,\dot b$ is anti symmetric wrt $t,n,b$. But is there any geometric intuition that immediately gives this ...
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28 views

Cohomology of two pieces of torus

In an exercise from an old exam, I found myself confronted with $M=\{(\sqrt{x^2+y^2}-2)^2+z^2=1\}$, $U=M\cap\{x\neq0\vee y>0\}$, $V=M\cap\{x\neq0\vee y<0\}$ and $U\cap V$, all subsets of ...
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Killing vector field along a geodesic

I was trying to show that a Killing vector field satisfies the Jacobi Equation for a geodesic, just by assuming that \begin{equation} \nabla_\mu X_\nu + \nabla_\nu X_\mu=0 \end{equation} Indeed, if I ...
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59 views

Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$

I'm lost on solving the following problem. Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$, where $\theta$ is the angle between the positive $x$-axis and the ...
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1answer
64 views

Metric on n-sphere in terms of stereographic projection coordinates

The metric on the $n$-sphere is the metric induced from the ambient Euclidean metric. Find the metric, $d\Omega^2_n$, on the $n$-sphere and the volume form, $\Omega_{S_n}$ , of $S^n$ in terms of the ...
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31 views

How to show that two vector fields commute?

Could anyone help me with how to start to solve the following problem? From this problem as well as this, I have: Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such ...
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18 views

Find $d\left(\frac{\partial\left(x,y\right)}{\partial\left(\delta_1,\delta_2\right)}\right)$ with the exterior product

Let $J_{\delta_1,\delta_2}^{x,y}$ denote the Jacobian $\partial\left(x,y\right)/\partial\left(\delta_1,\delta_2\right)$. Suppose I wanted to find $d\left(J_{\delta_1,\delta_2}^{x,y}\right)$ ...
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33 views

A diffeomorphism whose tangent map preserves dot products is an isometry.

I'm having trouble solving the following problem. If $F:\mathbb{R^3} \to \mathbb{R^3}$ is a diffeomorphism such that $F\ast$(the tangent map of $F$) preserves dot products, show that $F$ is an ...
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1answer
60 views

$\psi ( \frac {\pi}{2}, \frac {\pi}{6})$ and calculating problems? [closed]

I ran into a problem, $u=\psi (x,t)$ be a solution of partial deferential equation with following condition on boundary, how we reach the value of $\psi ( \frac {\pi}{2}, \frac {\pi}{6})$? ...
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48 views

Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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19 views

Proof that any unit-speed-reparametrization of a curve preserves orientation and is an inverse of an arc length function based at some $t_0$.

I am not able to prove the following two facts about a unit speed reparametrization of a curve. Let $\alpha$ be defined on some interval $I$ and define for $t_0\in I$ ...
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28 views

An orthogonal transformation with determinant 1 rotate $\mathbb{R^3}$ around an axis.

I'm studying differential geometry and need confirmation on the solution of the following problem. A rotation is an orthogonal transformation $C$ such that det $C=+1$. Prove that $C$ does, in fact, ...
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43 views

How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for ...
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3answers
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is every totally geodesic submanifold the set of fixed points of some isometries?

It is well known that the set of fixed points of an isometry $\phi:(M,g)\rightarrow (M,g)$ is a totally geodesic embedded submanifold. (e.g here ). I ask whether the converse is true, i.e is every ...
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41 views

immersion of punctured torus in plane

Let $S^1 \times S^1$ be the $2$-torus. If a point $a=(p,q)$ of the torus is removed, i.e., it is punctured at one point then how can I show that it can be immersed in the plane, i.e., in $\Bbb{R}^2$? ...
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327 views

Unit speed reparametrization of curve

I am learning Elementary Differential Geometry by O'Neill and having a hard time with this exercise. Suppose that $\beta_1$ and $\beta_2$ are unit-speed reparametrizations of the same curve $\alpha$. ...
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Associated bundle - valued 1 forms.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Fact: (as can be found in lecture notes here) Functions ...
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Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
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Jets and vertical differential

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: ...
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34 views

Vector fields that are smooth on the open unit ball $B$, are smooth on $\mathbb{R}^n$ if they are zero outside $B$

Could anyone help a little with the following problem? It is a continuation of this problem, but I will restate the things that are needed: Fix $\varepsilon \in (0, 1)$ and choose a smooth ...
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3answers
41 views

Derivative of a function in $\mathbb{R}^n$

Let $f:\mathbb{R}^m\to\mathbb{R}^m$ be a differential function. Let $Df(x)$ be the derivative of f at $x\in\mathbb{R}^m$. Which of the following is/are correct? $Df(0)(u)=0 \forall u\in\mathbb{R}^m$ ...
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Existence of a differentiable function given a unit gradient field

I'm trying to prove that "Given a unit vector field $V$, it can always be uniquely determined a differentiable function $f$ that satisfies $\nabla f = V$." To provide you more information, the unit ...
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756 views

Spivak and Invariance of Domain

On p.3 of the first volume of Spivak's Comprehensive Introduction to Differential Geometry, he says that it is an "easy exercise" to show that the invariance of domain theorem (if ...
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460 views

Equation of a straight line in spherical coordinates

I'm trying to prove the angle sum formula for a triangle on the surface of a sphere. In order to do this I wanted to create a general triangle on the sphere, with one vertex at $\theta = 0$ and one ...
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1answer
24 views

Basic properties of smooth curves

Suppose $\Gamma$ is simple smooth closed curve parametrized by $\gamma:[0,1]\to\Gamma.$ Let $$\gamma(t)-\gamma(s)=(t-s)F(t,s)\,\,\,\,t,s\in[0,1].$$ Can we conclude that $|F(t,s)|>0$ for all ...
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107 views

Transversality of Vector Fields Defined in terms of Diff. Forms and Open Books.

All: Sorry for the length of the post, but I think it is necessary to set things up so that the post is understandable. I'm trying to understand how it is that the transversality (in this case , the ...
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What is some elementary differential geometry textbook that is self contained and are intermediate level?

What are some intermediate differential geometry textbook that are more advanced than pressley's, Barrett's, Christian's and krezig's books and are self contained but below the level spivak's vol ...
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1answer
73 views

Proving a certain function is injective

I have found the following exercise on an exam for Geometry three dating to a past year. Let $F(u,v)=((2-v\sin\frac{u}{2})\sin u,(2-v\sin\frac{u}{2})\cos u,v\cos\frac{u}{2})$, with ...
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The meaning of variables and derivations in Souriau's book

As far as I see, Souriau is using unconventional notions in his book "Structure of Dynamical Systems". He explains these notions in §2. of Chapter I, but it is a puzzle for me. Mainly because he ...
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38 views

How to proof that bracket of two vector field can be computed by second derivation

Can some one give a hint how can I proof that where $\phi$ indicated the flow of vector fields.
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28 views

Green's theorem via Stokes's theorem

I am considering the following form of Stokes's theorem: Let $\omega$ be an $n-1$ differential form with compact support on an oriented manifold of dimension $n$. Let us consider the boundary ...
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Pullback 1-differential form

Let $(x_0,x'_0) \in \mathbb{R}^2$ be initial data for the Euler Lagrange equation with some given Lagrangian $L: \mathbb{R}^2 \rightarrow \mathbb{R}.$ Then $F^t$ is the flow that maps the initial ...
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27 views

Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can ...
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1answer
48 views

Compute Euler characteristic of a compact manifold

We have the manifold embedded in $\mathbb{R}^4$ given by $$M=\{(x,y,z,w)|2x^2+2=2z^2+w^2,3x^2+y^2=z^2+w^2\}$$ How could I compute the Euler characteristic? I've no idea computing the homology group of ...
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30 views

Why not every homogeneous manifold is parallelizable?

It is obvious that not every homogeneous manifold is parallelizable (take for example the two-sphere $S^{2}$). In contrast, every Lie group $G$ is parallelizable, as you can construct a pointwise ...
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Osculating Circle in Differential Calculus

I am working with an osculating circle as the curve of closest contact to a curve in differential calculus and my book takes some confusing steps that I do not understand. It says: Let $f(x)$ be the ...
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Question about parallel displacement on a surface

This is Problem 9.6(1) from the book The Geometry of Physics: What's wrong with the following argument? A vector $\mathbf v$ is parallel displaced around a small closed curve $C = ...
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How to prove that the tangent map $T\phi$ into the pullback bundle is smooth?

Assume $\phi: M\rightarrow N$ is smooth. Let $\phi^*(TN)$ be the pullback bundle of $TN$ by $\phi$. Define $T\phi:TM\rightarrow \phi^*(TN)$ as follows: $T\phi(m,v)=(m,d\phi_m(v)) $. We also have the ...
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A “Theorem Style” Problem Book in Differential Geometry

I am trying to teach myself differential geometry using Lee's Introduction to Smooth Manifolds. To test my understanding, and learn the subject better, I am looking for a good problem book in ...
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40 views

Proving that these curves intersect

Let $\Gamma$, $\Sigma$ be two curves with ranges in $(\{0\}\cup\mathbb{R}_{+})^2$. $\Gamma$ starts on the $y$ and ends on the $x$ axis: $\Gamma(0)=(0,\gamma_2),\Gamma(1)=(\gamma_1,0)$. $\Sigma$ is a ...
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2answers
31 views

Möbius strip as a non-trivial principal bundle

There is a well-known theorem that a principal bundle is trivial if and only if it admits a global section. I'm trying to get a good picture of what this theorem means. The Möbius Strip can be ...
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18 views

Twice differentiable functions that are harmonic

This is a question that I have spotted in a textbook for differential geometry. Determine all twice differentiable non-zero functions g : R $\rightarrow$ R and h : R $\rightarrow$ R such that $f ...
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1answer
66 views

Key differences between almost complex manifolds and complex manifolds

I know the technical difference between an almost complex manifold and a complex manifold, namely in the former the almost complex structure $J$ may not be integrable while in the later it is. ...
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28 views

On critical values of a linear function $g:SO_2\times \mathbb R^2\to \mathbb R^2$

Consider map $g:SO_2\times \mathbb R^2\to \mathbb R^2$, $g(A,v)=Av$, where $A\in SO_2$ is an orthogonal $2\times 2$ matrix and $v\in \mathbb R^2$ is a $2$-vector. Show that $0$ is a critical value. ...
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Unbounded Geodesics and Nonpositive Curvature

I have the following interesting(?) question: Let $M$ be a complete Riemann manifold. Suppose that it is of non-negative curvature. I want to know if $M$ has unbounded geodesics. As the question is ...