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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
33 views

Normal coordinates and the metric matrix

While trying to follow and check the proof of Theorem 1 in this work on manifold averaging I reached the notion of normal coordinates. An important property is that the metric tensor at a point ...
1
vote
1answer
44 views

Condition to be conformal

I am looking at the following exercise: Let $\Phi : U \rightarrow V$ be a diffeomorphism between open subsets of $\mathbb{R}^2$. Write $$\Phi (u, v)=(f(u, v), g(u, v))$$ where $f$ and $g$ are ...
1
vote
0answers
39 views

Connection with zero torsion and curvature

I remember encountering the following theorem many, many time ago: if a connection $\nabla$ has $R = 0$ and $T = 0$ then... The problem is that I cannot remember the conclusion. Was it that the ...
1
vote
0answers
29 views

Is the curl of a vectorfield expressable in terms of a Lie derivative?

I am loving differential forms and the general version of Stokes theorem. However, I am also fond of the intuitive pictures of my older physics days of $grad$, $div$ and $curl$. I recently stumbled ...
2
votes
1answer
44 views

How to express the property “spiraling around” in differential geometry

I am starting to learn differential geometry, and reading the book "Differential Geometry of Curves and Surfaces" of Manfredo. The I got stuck on this problem: Let $\alpha(t) = ...
1
vote
1answer
29 views

Approximation of piecewise smooth curves with same-lenght smooth curves in Riemannian manifolds

Let $M$ be a Riemannian manifold, and let $\gamma : [a,b]\to M $ be a piecewise smooth curve. Then, using Whitney's theorems, it can be proved that $\gamma$ is homotopic (by a homotopy relative to $a$ ...
1
vote
0answers
31 views

Diffusion on a Boundaryless Manifold and Tesselation

Suppose we are dealing with diffusion on a boundaryless manifold $M$. When people use the finite-element method to find an approximate solution, I always see them use \begin{align} \int_{M^*} W \Delta ...
0
votes
1answer
29 views

How to understand it will sweep out a 2-dim manifold?

As red line in picture below, I really can't imagine why it will sweep out a 2-dim submanifold. Below picture is from the 9th page of John M. Lee's 'Riemannian Manifolds: An Introduction to ...
0
votes
1answer
41 views

The parameter curves are asymptotic curves

I am looking at the following exercise: Let $p$ be a hyperbolic point of a surface $S$. Show that there is a patch of $S$ containing $p$ whose parameter curves are asymptotic curves. Show that ...
6
votes
3answers
181 views

For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
2
votes
1answer
35 views

Polyakov action in complex coordinates

Let $\Sigma$ be a compact $2$-manifold with riemannian metric $g$ and $f:\Sigma \to \mathbf{R}^n$ given locally by $f_1(x_1,x_2),\dots,f_n(x_1,x_2)$. Define $$ S(f,g) = ...
2
votes
1answer
67 views

Is this a misprint in Do Carmo's 'Curves and Surfaces'?

I'm reading the following section from the book 'Curves and Surfaces' by Do Carmo, but I'm stuck and after having gone over this like 10 times I'm starting to think it must be a misprint. The problem ...
1
vote
2answers
65 views

Which is the intersection?

I am looking at the last question of the following exercise: $$$$ Which exactly is the intersection of any surface from one family of the triply orthogonal system with any surface from another ...
2
votes
1answer
122 views

Why do these two facts imply that $S^n$ is not parallelizable for $n$ even, $n \ge 2$?

Consider the following two facts. If $S^n$ admits a vector field which is nowhere zero, the identity map of $S^n$ is homotopic to the antipodal map. For $n$ even, the antipodal map of $S^n$ is ...
1
vote
0answers
34 views

Is this subset of $\mathbb{R}^{3}$ a topological manifold?

Consider the set $\mathcal{M}_{1} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ y = -1 \ \}$, this is a plane. Also consider the set $\mathcal{M}_{2} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ x=y=0 \ \}$, which ...
2
votes
1answer
26 views

$X_z=\frac{d}{dt}_{|t=0} \Phi_t(z)$ has flow $\Phi_t$

Let $M$ be a manifold, $\Phi_t, t\in \mathbb R$ a one parameter group of diffeomorphisms and $X$ a vector field on $M$ definied by $$X_z:=\frac{d}{dt}_{|t=0} \Phi_t(z).$$ Show that $\Phi_t$ is the ...
8
votes
1answer
117 views

Is a general smooth rescaling of a complete vector field itself complete?

$\newcommand{\Ga}{\Gamma}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\til}{\tilde}$ $\newcommand{\M}{M}$ $\newcommand{\ep}{\epsilon}$ $\newcommand{\brk}[1]{\left(#1\right)} $ ...
3
votes
2answers
79 views

$S^n$ admitting nowhere zero vector field implies identity map of $S^n$ is homotopic to antipodal map? [closed]

If $S^n$ admits a vector field which is nowhere zero, does it follow that the identity map of $S^n$ is homotopic to the antipodal map?
0
votes
1answer
26 views

Integral of a portion of a curve

I'm struggling with this question: It says: let $C$ be the portion of the curve $y=2 \sqrt{x} $ between $(1,2)$ and $ (9,6).$ Find $ \int_C3y \, ds$ Any clue that would assist me?
0
votes
1answer
43 views

How does the solution of a differential equation on a manifold yield a map?

In: "A solution $x^μ(λ)$ is a map from $\mathbb{R} → M$": Why is $x^μ(λ)$ considered a map and why does it go from $\mathbb{R} → M$? I can't seem to illustrate this in my mind. In:"If the manifold ...
3
votes
1answer
42 views

Is parallelizability equivalent to the set of vector fields being free?

We have the $C^{\infty}(M)$-module $\mathcal{D}^1(M)$ of vector fields over a $C^{\infty}$ manifold $M$. Is being parallelizable equivalent to this module being free, of dimension $n$? I have the ...
1
vote
1answer
33 views

When are two vector fields $C^1$ close?

For simplicity let us assume we have a fixed compact manifold M. Introduction: While considering $C^\infty(M,\mathbb{R})$ one can say that two functions $f,g$ are $\epsilon-close$ iff for fixed ...
1
vote
0answers
10 views

Unique function that gives an angle to a point of an interval.

(Sorry for my poor English in advance, as it is not my first language. Sorry for the vague title too, as I didn't know how to summarize the topic of the question). Here is an exercise that was ...
0
votes
0answers
28 views

Orientation preserving and orientation reserving of a parameterisation of a curve

The definition states: Let $\mathbf{x}:\left [ a,b \right ]\rightarrow \mathbb{R}^{n}$ be a piecewise $\mathit{C}$ path. We say that another $\mathit{C}$ path $\mathbf{y}:\left [ c,d \right ...
1
vote
1answer
56 views

Prove S is a manifold.

At the moment the definition of a manifold I'm working with is that of a set $X$ equipped with a smooth atlas $A$. I want to prove that $\{(a,b)\in \mathbb{R}^n\times\mathbb{R}^n \mid a\cdot a=b \cdot ...
0
votes
0answers
17 views

Parallely transported vector continuous with respect to homotopy of curve?

Let $M$ be a smooth manifold with connection. Choose two points $p,q \in M$ and connect them with curve $\gamma _0$. Suppose I take fixed vector at $x$ and parallel transport it to $y$. Denote this ...
2
votes
1answer
55 views

Best Fit Line with 3d Points

Okay, I need to develop an alorithm to take a collection of 3d points with x,y,and z components and find a line of best fit. I found a commonly referenced item from Geometric Tools but there doesn't ...
0
votes
0answers
26 views

To find surface and curve with given normal & geodesic curvatures, geodesic torsion.

Find a surface parametrization if normal/tangential curvatures and geodesic torsion are given by: $$ \kappa = a \cos s, \tau = b \sin s , \tau_g = c, $$ if the surface contains point (0,0,0), ...
1
vote
0answers
26 views

“Scattering” in Riemannian spaces.

Let $(M,g)$ be a compact $n$-dimensional Riemannian space with boundary $\partial M$. Consider different types of "scattering" functions: 1) \begin{equation} y:TM \to \partial M: (x,v_x) \mapsto ...
1
vote
1answer
28 views

Schaum's Differential Geometry exercise on curvature

Page 72 exercise 4.5, there is the following situation: There is a curve $\underline{x}(t)$ with $t$ not a natural parameter. I have to find the curvature vector $\underline{k}$ and the curvature $k$ ...
0
votes
3answers
60 views

Orientable surface

Suppose that two smooth surfaces $S$ and $\tilde{S}$ are diffeomorphic and that $S$ is orientable. I want to prove that $\tilde{S}$ is orientable. $$$$ Since $S$ and $\tilde{S}$ are ...
0
votes
1answer
23 views

Convert to cartesian?

How would I convert $X(t)=\cos(t)a+\sin(t)b$ to cartesian, where $a=(3,3)$ and $b=(-1,1)$. I tried saying $x(t)=3\cos(t)-\sin(t)$ and $y(t)=3\cos(t)+\sin(t)$ but I am stuck on how to remove the $t$.
3
votes
1answer
44 views

Inward/Outward-pointing tangent vector is well-defined

Let $M$ be a smooth manifold with boundary and $p\in \partial M$. We say a tangent vector $v\in T_pM$ is inward-pointing if in a chart $x$ with $v=v^i\partial/\partial x^i$ (using the summation ...
3
votes
2answers
36 views

parallel postulate of Euclidean geometry and curvature

In elementary geometry, we have two standard examples which violate the (strong) parallel postulate of Euclidean geometry: in hyperbolic geometry, we have more than one parallel through a point which ...
2
votes
1answer
30 views

The two corollaries of Stoke's theorem

The two corollaries of Stoke's theorem is as follows: 1) $\int \left ( \vec{\nabla}\times \vec{v} \right ).d\vec{a}$ depends only on the boundary line, not on the particular surface used 2 $\oint ...
1
vote
2answers
34 views

Radius of circle that lies on the surface of a ball

If I intersect a ball with a plane, then I can see how to make sense of the radius of the resulting circle of intersection. But if the circle lies strictly on the surface of the ball (sphere), then ...
5
votes
3answers
443 views

Is the metric on the circle, induced from the plane, not a flat one?

My question concerns the highlighted part posted below, from Wikipedia article. (Link to the revision at the time of this post.) I'd say I can't detect the curvature of the unit circle if I go along ...
0
votes
1answer
56 views

How can we show that parallel transport is invertible?

We have that the map $\Pi^{pq}_{\gamma}: T_pS \rightarrow T_qS$ that takes $v_0 \in T_pS$ to $v_1 \in T_qS$ is called parallel transport from $p$ to $q$ along $\gamma$. $\Pi^{pq}_{\gamma}: T_pS ...
0
votes
1answer
43 views

Proof of the Envelope theorem for parametric functions in the Cartesian Plane

Given a family of parametric functions in the Cartesian Plane $(f(t,k);g(t,k))$, their envelope is found by solving the equation: $\frac{\partial f}{\partial t}\frac{\partial g}{\partial ...
0
votes
1answer
34 views

Is there coordinates $u,v$ such that $e^{-x^2-4y^2}dx\wedge dy = du\wedge dv$?

Is there coordinates $u,v$ on $\mathbb{R}^2$ such that $e^{-x^2-4y^2}dx\wedge dy = du\wedge dv$?
1
vote
0answers
15 views

Weyl Transformations and Group actions

I have the following question. Let $(M,g_{ab})$ be a Riemannian manifold $M$ with metric $g$, and with an action of a Lie group $G$. Moreover, the Riemannian metric $g_{ab}$ is taken to be invariant ...
26
votes
1answer
343 views

“the only odd dimensional spheres with a unique smooth structure are $S^1$, $S^3$, $S^5$, $S^{61}$”

This (long) paper, Guozhen Wang, Zhouli Xu. "On the uniqueness of the smooth structure of the 61-sphere." arXiv:1601.02184 [math.AT]. proves that the only odd dimensional spheres with a ...
1
vote
1answer
37 views

Is the $\alpha$-curve a topological submanifold of $\mathbb{R}^2$

Consider the following subset of $\mathbb{R}^2$ defined by $\mathcal{M}=\{ \ (x,y)\in\mathbb{R}^2\ |\ y^2=x^2(x+1)\ \}$. I'm supposed to decide whether or not $\mathcal{M}$ is a topological ...
0
votes
1answer
51 views

Geodesic - How can we continue?

I want to show that a geodesic with nowhere vanishing curvature is a plane curve if and only if it is a line of curvature. I have done the following: Let $\gamma$ be a geodesic with nowhere ...
0
votes
0answers
103 views

Taylor expansion - How can we deduce that?

We have that $$\tilde{E}=\frac{u^2}{r^2}+\frac{Gv^2}{r^4}, \ \ \tilde{F}=\left (1-\frac{G}{r^2}\right )\frac{uv}{r^2}, \ \ \tilde{G}=\frac{v^2}{r^2}+\frac{Gu^2}{r^4}$$ and that ...
1
vote
1answer
27 views

Compact surface - Tube/Torus

How can we explain, without giving a detailed proof, why the tube $$\sigma (s,\theta )=\gamma (s)+a(n(s)\cos\theta+b(s)\sin\theta )$$ (where $n$ is the principal normal of the curve $\gamma$ and $b$ ...
12
votes
3answers
275 views

What's true in $\mathbb{R}^4$, false in $\mathbb{R}^3$ and uninteresting in $\mathbb{R}^5$?

What are some interesting and easy-to-understand (for non-differential geometers) facts about subobjects of $\mathbb{R}^4$ that are not only false in $\mathbb{R}^3$, but also specific to the structure ...
0
votes
0answers
28 views

Orthogonal projection onto the tangent space

Let $\tilde{\gamma}$ be a reparametrization of $\gamma$, so that $\tilde{\gamma}(t) = \gamma (\phi (t))$ for some smooth function $\phi$ with $\frac{d\phi}{dt} = 0$ for all values of $t$. If $v$ is ...
2
votes
1answer
30 views

Distance between the geodesic and the $z$-axis

Let $\gamma$ be a unit-speed curve on the helicoid $$\sigma (u,v)=(u\cos v, u\sin v, v)$$ I have shown that $$\dot u^2+(1+u^2)\dot v^2=1$$ and that if $\gamma$ is a geodesic on $\sigma$ then $$\dot ...
2
votes
1answer
166 views

Parallels and meridians

Consider the surface patches $$\sigma (u, v) = (\text{cosh } u \cos v, \text{cosh } u \sin v, u), \ \ \tilde{\sigma}(u, v)=(u \cos v, u \sin v, v)$$ parametrizing a catenoid and a helicoid, ...