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)

1
vote
0answers
24 views

Showing that vector field $\mathbb{Z}$ satisfies $\mathbb{Z}\cdot(\nabla \times \mathbb{Z})=0$,connected to the Frobenius Theorem

Suppose $\mathbb{Z}$ is a smooth vector field on $\mathbb{R}^3$ with $\mathbb{Z}^3(x,y,z) \neq 0$. a) Find functions $f$ and $g$ such that the vector fields $\mathbb{X}=(1,0,-f)$ and ...
0
votes
0answers
13 views

Another Problem integrating when attempting a solution with the Poincaré Lemma

a) I think that the answer should be $d\nu=10z dx \wedge dy \wedge dz$ b) and c) are easy. d) This is part I am having troubles with. $\begin{align} i_{\hat{\mathbb{X}}_t}\beta &= ...
3
votes
2answers
74 views

Does $\omega \wedge \mathrm{d} \omega=0$ (where $\omega$ is a non-vanishing $1$-form) imply $\mathrm{d} \omega \in < \omega >$?

Let $\omega$ be a non-vanishing (for clarification: nowhere vanishing) smooth $1$-form on a smooth manifold $M$, if $\mathrm{d}\omega \wedge \omega =0$, do we already have $\mathrm{d}\omega= \sum a_i ...
4
votes
1answer
23 views

What can you say about injection, immersion, embedding for the torus?

Define $\varphi_a: \mathbb{R} \to T$ where $T = S^1 \times S^1$ is the torus via$$\varphi_a(x) = (e(x), e(ax)),\text{ }e(x):=e^{2\pi i x}$$and $a > 0$ is some parameter. Determine for which $a$ ...
7
votes
3answers
183 views

“Drawable” Examples of Vector Bundles

I'm looking for examples of vector bundles that can be easily drawn or "illustrated" on a whiteboard for a talk I am giving. I know of a couple simple examples that I could use: When our base ...
4
votes
1answer
17 views

uniform continuity, differentials

Let $\{f_n\}_{n=1}^\infty$ be a sequence in $C^1(U)$ where $U \subset \mathbb{R}^d$ is open. Suppose $f_n \to f$ uniformly on compact subsets of $U$. Assume further that $df_n \to A$ in the same sense ...
0
votes
0answers
27 views

Using Frobenius' Theorem for 3 functions in 2 variables [on hold]

i) 1) $v= \frac{\partial u}{\partial x}$ 2) $w= \frac{\partial u}{\partial t}$ 3) $\frac{\partial v}{\partial t}= \frac{\partial w}{\partial x}$ 4) $\frac{\partial v}{\partial x}= \frac{\partial ...
1
vote
1answer
26 views

Vector fields and plateau functions.

Let $M$ a differentiable manifold and $X$ a vector field on $M$. Given $m \in M$, there exists $(W; x^1,...,x^n)$ a coordinate system around $m$. In $W$, the vector field $X$ can be written as $\sum ...
0
votes
1answer
50 views
+50

Problem integrating in problem using the Poincaré Lemma

a) It is easy to show that $d\beta=0$. b) $\begin{align}\hat{\mathbb{X}}_t &= (\frac{\partial}{\partial t}\hat{\Phi}_t) \hat{\Phi}_t^{-1}) \\ &= (\frac{\partial}{\partial t}\hat{\Phi}_t) ...
4
votes
1answer
44 views

relation between first fundamental form for different parametrization

The sphere has a parameterization map for a surface patch $\phi(u,v)=(u,v,\sqrt{1-u^2-v^2})$. It has another parametrization map for a surface patch $\beta (x,y)=(\sin x \cos y,\sin x \sin y,\cos ...
1
vote
1answer
22 views

Showing that $F^*(L_{\mathbb{Y}}\omega)=L_{\mathbb{X}}(F^*\omega)$

Let $F:U \rightarrow V$ be a diffeomorphism between open sets in $\mathbb{R}^n$. Let $\mathbb{Y}$ be a vector field on $V$ and $\omega$ a $k$-form on $V$. Show that ...
3
votes
0answers
64 views

Maximum diameter for the preimage of a point when the degree is not 1 or -1

When the degree of a map $S^n \rightarrow S^n$ isn't 1 or -1, are there always two points that map to the same point and whose distance from each other can be bounded below by a function of the ...
0
votes
1answer
27 views

Can a map from a p-simplex to the n-sphere be surjective?

If $p<n$, is this possible? I am confused about this. I am trying to prove that the i-th singular homotopy group of the n-sphere is a subset of the i-th homotopy group of $\mathbb{R}^n$ but I am ...
0
votes
0answers
33 views

Pull-back of a one-form on a sphere.

Let $\iota: S^2 \to \mathbb{R}^3$ be the inclusion map and choose a chart $(U,f)$ on $S^2$, where $U=\{(x,y,z)\in \mathbb{R}^3: z>0\}$ and $$f: U \to \mathbb{R}^2,$$ $$ (x,y,z)\mapsto (x,y). $$ I ...
4
votes
0answers
33 views

The missing vector derivative operation

Generally, the differential operator $d$ takes differential $k$-forms to $(k+1)$-forms. In $\mathbb{R}^n$ (or more generally on an oriented Riemannian manifold), we can identify vector fields with ...
1
vote
1answer
40 views

Module of differentials in the functorial approach to schemes and quasi-coherent modules

Recall that for a functor $X : \mathsf{CAlg}(R) \to \mathsf{Set}$ from commutative $R$-algebras to sets one can define quasi-coherent $\mathcal{O}_X$-modules as "compatible" families of $A$-modules ...
2
votes
1answer
20 views

Is this cylinder a regular surface?

Let $C$ be a figure $‘‘8"$ in the $xy$ plane and let $S$ be the cylinder surface over $C$; that is, $$S=\{(x,y,z)\in\mathbb{R^3}:(x,y) \in C \}$$ Is the set $S$ a regular surface? I know that the ...
5
votes
0answers
79 views

How important is Differential Geometry for Number Theory?

The title pretty much says it. To elaborate slightly, I am, of course, aware of the huge role played by Algebraic Geometry in Number Theory but I'm not so sure about Differential Geometry. I would be ...
3
votes
1answer
90 views
+50

Describing non-vanishing $1$-forms on two dimensional manifolds.

Let $h_1 \mathrm{d}x_1 + h_2 \mathrm{d}x_2$ be a non-vanishing $1$-form on a $2$-dimensional manifold. Why do locally exist smooth functions $f,g$ with $f\mathrm{d}g= h_1 \mathrm{d}x_1 + h_2 ...
1
vote
1answer
78 views

Moving frame in a semi-Riemannian manifold

Can someone point me some reference for the moving frame theory in semi-Riemannian manifolds, using differential forms? In special, I'm looking for a version of Cartan's structural equations. I've ...
1
vote
1answer
26 views

Show that $i_Yi_Xd\omega=d\omega(X,Y)$ for $\omega$ a $1$-form

If $\omega$ is a $1$-form, how does $i_Yi_Xd\omega=d\omega(X,Y)$? I get that $d\omega$ is a 2-form. So $i_X(d\omega)=d\omega(X,v_{2})$. So how do we proceed? I dont see how the step ...
0
votes
1answer
24 views

Roulette (curve) parameterization

I was wondering about the parameterization of a roulette on Wikipedia. A roulette is a curve formed by a point associated to one curve as it rolls upon another fixed curve. Wikipedia says, if $f$ is ...
1
vote
3answers
53 views

Economically computing $d\beta$

$\displaystyle \beta = z\frac{x dy \wedge dz + y dz \wedge dx + z dx \wedge dy}{(x^2+y^2+z^2)^{2}}$ Show that $d\beta=0$. So, let $r=x^2+y^2+z^2$, $\begin{align} \displaystyle d\beta &= ...
2
votes
1answer
29 views

Equivalence of unoriented knots by ambient isotopy

I'm trying to understand the equivalence of unoriented knots in oriented 3-manifolds for my thesis, and getting confused. I have not found a satisfactory definition of this equivalence. My ...
3
votes
1answer
21 views

Given two local parameterizations and corresponding fundamental forms, find a $2\times 2$ matrix that equates them.

Horridly written title, but please see the image below for the problem statement: I'm not sure how to use the chain rule to express the first differentials in terms of the second. I know the answer ...
1
vote
2answers
53 views

Is a manifold orientable if it has an atlas which has negative determinants for all of its transition maps?

I've tried to construct a non-vanishing n-form on a manifold given this condition, like one for an orientable Riemannian manifold. However, the partition of unity won't work; there's a change of sign ...
0
votes
0answers
49 views

What geometric shape is “the perfect milkshake container”?

A sphere is the best shape for a snowball if you want to maximize the amount of time before the snowball melts. This is because the ratio of the surface area divided by the volume is the smallest. ...
2
votes
0answers
70 views

Showing a property of a curvature tensor in $S^2$

Consider $S^2 \subset \mathbb{R}^3$. I need to show that if $$R_{ijkl} = -g(R(\partial_i,\partial_j)\partial_k,\partial_l)$$ is a curvature tensor in $S^2$ and $g$ is a metric also in $S^2$, then ...
1
vote
1answer
36 views

Tangent Space: Identifications

Given a manifold $M$. Denote a chart by $\kappa$. Introduce the directional derivative: $$\partial:\mathbb{R}^n_a\to T_a\mathbb{R}^n:v\mapsto\partial_v\rvert_a$$ That is an isomorphism with inverse ...
0
votes
1answer
41 views

Integrate the gaussian curvature

Let $T$ be a torus. We have a parameterization by $((c+a \cdot cos(v))cos(u),(c+a\cdot cos(v),a\cdot sin(v))$ for $u,v \in [0,2\pi)$. The first fundamental form is given by $E=(c+a\cdot cos(v))^{2}, ...
0
votes
2answers
49 views

Seamlessly connect a sine curve and a parabola

I want to seamlessly connect an unknown parabola to a known sine wave. The equations are: s(x) = a sin(bx + c) p(x) = Ax^2 + Bx + C I want to draw ...
5
votes
1answer
30 views

Exists open subset and one-to-one $C^1$ mapping such that mapping of intersection is open subset

Let $M$ be a smooth $k$-manifold in $\mathbb{R}^n$. Given ${\bf p} \in M$, how would I go about showing there exists an open subsets $W$ of $\mathbb{R}^n$ with ${\bf p} \in W$, and a one-to-one $C^1$ ...
0
votes
0answers
11 views

Surfaces swept out by trihedron vectors

Surfaces swept out by unit tangent of a curve on a surface is developable. Are normal and bi-normal swept out surfaces also developable?
2
votes
0answers
40 views

How to phrase this identity in differential form language?

If the vector field $\mathbf B$ on $\mathbb{R}^3$ is constant, then the vector field $$ \mathbf A = \frac 1 2 \mathbf B \times \mathbf r $$ satisfies $$ \nabla \times \mathbf A = \mathbf B. $$ This ...
1
vote
1answer
60 views

Manifold which is union of two balls is topologically a sphere

In Petersen's book while proving sphere theorem the following fact has been stated without prove : Let $M$ be a connected $n$ dimensional smooth manifold such that $M=B_{1}\cup B_{2}$ where $B_{i}$'s ...
0
votes
0answers
17 views

Variation of geodesic and Jacobi field

I was reading on Jacobi field of a geodesic, and noticed that given a geodesic $\gamma$ it is defined using the term of variation or family of geodesics $\gamma_s$ but never mentioned how to create ...
1
vote
1answer
27 views

Computing the first fundamental form

Let $X$ be a smooth surface in $\mathbb{R}^{3}$. I want to compute the first fundamental form of $X$. Assume that $X$ has 2 different local parameterizations $r_1$, $r_2$ (i.e. for $r_1$ there is a ...
1
vote
0answers
36 views

Poincaré Lemma problems and computing contractions in an economical way

Let $x=(A,B,C,D)$ be coordinates on $\mathbb{R}^4$. $\displaystyle \beta = \frac{(AdB-BdA)\wedge(dC \wedge dD)+(dA \wedge dB)\wedge(CdD-DdC)}{(A^2+B^2+C^2+D^2)^2}$ I would like to compute ...
0
votes
1answer
32 views

Show that $d\beta=0 \iff p=n/2$

Let $\beta$ be the $(n-1)$-form on $\mathbb{R}^n \setminus \{0\}$ given by $\displaystyle \beta = \sum_{i=1}^{n}(-1)^{i-1}\frac{x^i dx^1 \wedge dx^2 \wedge \dots \wedge \hat{dx^i} \wedge \dots ...
1
vote
1answer
35 views

Does this surface exist

Does anybody of you know if there is a surface with first fundamental form $(g_{ij}) = \operatorname{diag}(1, \cos^2(u))$ and second fundamental form $(h_{ij}) = \operatorname{diag}(1, \sin^2(u))$? ...
0
votes
1answer
32 views

Why are there more complex than smooth structures?

I've read that given a topological manifold, there are only finitely many smooth structures on it (except for dimension 4) but many more (even uncountably many) complex structures. But doesnt this ...
0
votes
1answer
24 views

Curvature line parametrization

I have a question about the curvature line parametrization. We said that for a given surface $f: U \rightarrow \mathbb{R}^3$ we find a local curvature line parametrization such that both the first ...
2
votes
0answers
83 views
+50

Use of Poincare Lemma in solving $\nabla \times \textbf{A}(\textbf{r})=\frac{\textbf{r}}{r^3}$

Let $U = \mathbb{R}^3 \setminus \{(0,0,z) \}$ (ie $\mathbb{R}^3$ with the $z$-axis removed ) and consider $\beta$ on $U$ given by $\displaystyle \beta = \frac{x dy \wedge dz + y dz ...
0
votes
1answer
45 views

determine the principal curvatures of the surface defined as the tube around a space curve using the Frenet Serret frame.

Consider a regular unit speed curve $\alpha: (a,b) \to \Bbb R^3$. Then define the surface $S$ via the parametrization $x:(a,b)\times (-\pi,\pi)\to \Bbb R^3$ where $$x(u,v) = \alpha(t) + r(N(t)\cos ...
3
votes
1answer
48 views

Manifold with special cohomology group

I am trying to find if there is a orientable compact manifold $M$ of dimension 10 with the 5th cohomology group of De Rham $H^5_{DR}(M)\cong \mathbb R$. But, I can find such an example or prove that ...
3
votes
1answer
23 views

definition of differential does not depend on the choice of the curve, differential is a linear map

Let $\varphi: M \to N$ be a differentiable map. How do I show that the definition of the differential $d\varphi_p: T_pM \to T_{\varphi(p)}N$ of $\varphi$ at $p$ does not depend on the choice of the ...
3
votes
1answer
26 views

Every immersion is locally an embedding?

Let $\varphi: M \to N$ be an immersion and let $p$ be a point in $M$. How would I go about showing that there exists a neighborhood $V \subset M$ of $p$ such that the restriction $\varphi|V$ of ...
0
votes
1answer
59 views

Identity concerning Lie derivative of $k$-form $\omega$

Let $X$ and $Y$ be vector fields on $\mathbb{R}^n$. Show that for $\omega$, a $k$-form on $\mathbb{R}^n$, $(L_XL_Y-L_YL_X)\omega=L_{[X,Y]}\omega $. I try using Cartan's magic formula and get that ...
0
votes
0answers
46 views
+50

Identity of the pushforward of a vector field using a Jacobi bracket.

Let $Z(u,v)$ be the vector field $Z(u,v)=(u^2+u,v^2+v)$, let $\Gamma_t$ denote its flow. I have shown that $[X,Z]=Z-X$. Show that $(\Gamma_t)_*X=e^{-t}X-(e^{-t}-1)Z$. Could someone please show me ...
1
vote
1answer
23 views

Green's operator, differential forms

In "Foundations of Differential Manifolds and Lie Groups" by Frank Warner on page 225 there is defined Green's operator: $G: E^p(M) \rightarrow (H^p)^{\perp}$ by setting $G(\alpha)$ to equal the ...