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)

-2
votes
1answer
38 views

A function $\phi$ between two manifolds of class $C^\infty$ is constant if $d\phi$ [closed]

Let $M$ and $N$ two manifolds of $C^{\infty}$, $M$ connected, and $\phi:M\to N$ also of class $C^{\infty}$ so that in all point $m\in M$ the function $d\phi(m):M_m\to N_{\phi(m)}$ between the ...
2
votes
1answer
50 views

confusion on exercises from LEE's Book on Riemannian Manifold

I am reading John Lee's "Riemannian Manifolds an Introduction to Curvature" . At page 15 exercise 2.3 asks to prove that there exist a smooth extension of a function defined on a embedded ...
1
vote
2answers
37 views

Riemannian metrics on homogeneous spaces

Let G be a Lie group and H be a compact subgroup. The (left) coset space G/H is, up to an isomorphism, equivalent to the smooth homogeneous manifold M. My question is, is it possible to impose an ...
1
vote
1answer
27 views

Point of intersection of tangents

If the distance of two points $P$ and $Q$ from the focus of of a parabola $y^2 =4ax$ are $4$ & $9$ then what is the distance of the point of intersection of tangents at $P$ and $Q$ from the ...
0
votes
1answer
43 views

Local diffeomorpism is a covering?

$D:\tilde{M}\rightarrow \Omega$ is a local isometry (developing map) onto a connected manifold $\Omega$. $\tilde{M}$ is simply connected, has constant sectional curvature and there exists a deck group ...
2
votes
0answers
38 views

Classify the set $\{(x,y,z):f(x,y,z)=0, \nabla f=0\}$, where $f$ is a polynomial of degree at most 3.

Suppose that $f(x,y)$ is a nonzero polynomial of degree at most 2. Observe the following set: $$S=\{(x,y) : f(x,y)=0 ,\; \partial_{x}f(x,y)=0,\; \partial_{y}f(x,y)=0 \}.$$ Note that this set is the ...
1
vote
1answer
49 views

Stokes theorem for Cuboid

I need to proof stokes theorem $\int_Qd\omega=\int_{\partial Q}\omega\;$ for a 2-form and $Q\subset \mathbb R^3 \;$a cuboid. Since $\omega \;$ is a two form it can be written as $$\omega ...
0
votes
1answer
35 views

Differential of a smooth function on a manifold

Let $S^2$ be the sphere in $\mathbb{R}^3$, let's consider the (inverse) chart $\varphi$ $$x=\sin v\cos u, y=\sin v \sin u, z=\cos v$$ now let $f$ be the restriction of the linear aplication of ...
0
votes
2answers
23 views

Expressing tangent curve via level surface and graph of function

Given the sphere of radius $2$ centered at $(2,-1,0)$, find an equation for the plane tangent to it at the point $(1,0,\sqrt{2})$ in the following ways: 1) by considering the sphere as the graph of ...
1
vote
1answer
47 views

Differential of a form

I'm learning about Chern-Simons theory and my differential geometry is a bit rusty. The Chern-Simons 3-form is given by $\omega_3=tr(A\bigwedge\nolimits dA+\frac{2}{3}A\bigwedge\nolimits A ...
0
votes
1answer
29 views

Differential Geometry Proof Regarding Arclength, Tangents, Curvature, and Parameters

Consider a regular curve q(t) with arclength parameter s. Show that if $T(t_{n}) \neq T(t_{0})$ and $t_{n} \rightarrow t_{0}$, then $$1 = lim_{t_{n} \rightarrow t_{0}} \frac{|\theta(t_{n}) - ...
4
votes
2answers
62 views

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$?

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$? For starters, I know that if the $n$-dimensional $M$ can be immersed in ...
0
votes
1answer
27 views

Do Killing vector fields satisfy $\nabla_a X^a + \nabla_b X^b=0$?

Killing vector fields are those that verify $\mathcal{L}_X (g)=0$. This is equivalent to the following equation for a coordinate basis: $$\nabla_a X_b + \nabla_b X_a=0$$ Do Killing vector fields ...
1
vote
0answers
45 views

normal bundles are stably isotopic

I am searching for a reference about the following theorem: Let $M \xrightarrow{i_{0}} \mathbb{R}^{k},\,\,$ $M \xrightarrow{i_{1}} \mathbb{R}^{l}$ be two embeddings of the manifold $M$. We know by ...
0
votes
1answer
38 views

Addition in the space of orbits (under group action)

This question may be very trivial in this area, but I am a beginner to this and I stuck here. Could anyone please help me here! Let $\Gamma$ be a group whose identity is $e$. Let $X$ be a set and ...
1
vote
0answers
21 views

tangent vectors

Let $S$ be any regular surface in $R^3$ and let $p \in S$ be any point. From Classical differential Geometry I Know that the tangent space of $S$ at p is a subspace of $R^3$. If I see the surface ...
0
votes
1answer
49 views

Poisson bivector on the product of two manifolds

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by \begin{align} \pi_X = \sum_{i,j} ...
1
vote
0answers
38 views

Checking a proof involving flows

I am going through the proof of theorem 2.12 of the book Lectures on the geometry of Poisson manifolds by I. Vaisman. It's just a bit of differential geometry, but as I don't use these methods very ...
3
votes
1answer
61 views

The Pullback Bundle is an Embedded Submanifold of its Parent Space

$\newcommand{\mc}{\mathcal} \newcommand{\set}[1]{\{#1\}} \DeclareMathOperator{\pr}{pr} \newcommand{\at}{\big|} \DeclareMathOperator{\GL}{GL}$ Let $\pi:E\to N$ be a smooth vector bundle over a smooth ...
0
votes
0answers
19 views

Fiberwise isomorphism induces a bundle isomorphism

Given vector bundles $(\pi_1,E_1,M)$ and $(\pi_2,E_2,M)$ and a linear isomorphism defined in each fiber $f:E_p \rightarrow E_{f(p)}$, is it possible to define a $bundle$ isomorphism of the same vector ...
1
vote
0answers
12 views

Liegroup acting transitively => connected component acts locally transitive?

Suppose $G$ is a Liegroup acting transitively on a manifold $M$. Does that already imply, that the connected component $G^\circ$ of $G$ acts transitively on the connected components of $M$? I have a ...
1
vote
0answers
18 views

What is the most accessible reference on wall-crossing?

I am looking for a nice and easy to read reference on wall-crossing (in the context of Donaldson theory). Is there some accessible reference you have to suggest? I am interested in studying Donaldson ...
1
vote
1answer
54 views

Gaussian curvature expressed by torsion and curvature of its geodesics

Let $S$ be a minimal surface, and let $\gamma$ be a geodesic parametrized by arc length, with curvature $k$ and torsion $\tau$,show that in the points of $\gamma$ $$ -K=k^2+\tau^2, $$ where, $K$ is ...
0
votes
1answer
26 views

Why do I get infinity when I compute the Weingarten Map of the cone?

I am a newbie to differential geometry and I am learning on my own. In a practice problem, I tried to compute the Weingarten map for the standard cone parametrized as $ x(u, v) = (v \cos u ,v \sin u, ...
6
votes
5answers
750 views

Basic understanding of a metric.

What is a metric ? Do a metric depend on the system of coordinates I use ? Does it depend on surfaces (or higher dimensional manifolds. Correct me if I'm wrong using the word) the coordinate frames ...
0
votes
0answers
18 views

Non contable of homeomorphism family of the unitary open ball in itself

Let $M$ a variety of dimension $n$. Show that if have a structure of class $C^{\infty}$ then have a not countable number of such structures. entonces posee una cantidad no-numerable de tales ...
1
vote
1answer
12 views

Finding the image of a parameterised surface $S$

Let $\vec{X}=\langle s^2\cos(t),s^2\sin(t),s\rangle$ with $s\in [-3,3]$ and $t\in [0,2\pi]$ Find an equation for the image of $\vec{X}$ in the form $F(x,y,z)=0$ Finding myself lost here. ...
2
votes
1answer
62 views

Injective immersion that is not trajectory of any flow

Let $M$ be a compact manifold of dimension $m \geq 2$. Show that there exists an injective immersion of $\mathbb{R}$ in $M$, whose image is not the trajectory of any flow. I know how to do it for ...
0
votes
2answers
46 views

Tangent line of Lissajous curve?

I'm trying to find at how many points the tangent line of $(\cos(3t),\sin(2t))$ goes through the point $(3,0)$. My attempt: This is the same thing as saying for how many values of $t$ do we have ...
1
vote
0answers
31 views

Is exponential map an immersion?

Let $M$ be a connected Riemannian manifold. For $p\in M$, the injectivity radius at $p$ is the sup of the $\epsilon >0$ such that the Riemannian-distance ball $B_\epsilon (p)$ is a geodesic ball, ...
0
votes
2answers
43 views

Examples of surfaces

I have to find an example of a surface of revolution excluding a sphere and a cone. Is $\sigma(x,y)=(\cos x, 5, x^2+y^2)$ such an example? $$$$ I also have to find an example of a surface the ...
3
votes
1answer
137 views

Stiefel-Whitney class of complex projective spaces

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? Let $a_m$ be the maximal integer such that the $a_m$-th dual ...
2
votes
1answer
469 views

Exact definition of a circular helix

I know that a cylindrical helix is a curve $\alpha: I \to \Bbb R^3$, such that exists a constant vector $\bf u$ which makes a constant angle with the tangent vector $\bf T$ to the curve, at every ...
2
votes
1answer
76 views

How To Formalize the Fact that $(g, h)\mapsto dL_g|_h$ is smooth where $g, h\in G$ a Lie Group

Let $G$ be a Lie group. I am wondering if there is a way to say that the map $(g, h)\mapsto dL_g|_h$ defined on $G\times G$ is a smooth map (Here $L_g$ is the left translation map from $G$ to $G$ ...
8
votes
1answer
173 views

Helices in Lorentz-Minkowski space $\Bbb L^3$.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3$, with the metric: $$ds^2 = dx^2 + dy^2 - dz^2$$ (which I'll denote just by $\langle \cdot, \cdot \rangle$) If $\alpha$ is spacelike and ${\bf ...
5
votes
1answer
66 views

Lie groups where $x \mapsto x^2$ is a diffeomorphism?

In every Lie group $G$ the function $x \mapsto x^2$ is a local diffeomorphism in a neighbourhood of the identiy. (This is because its differential is: $v \mapsto 2v$ when considered as a map from ...
1
vote
0answers
39 views

Relation between differential geometry and differential geodesy

I am not exactly clear on what are the differences between differential geometry and differential geodesy. Are principles in differential geometry used in differential geodesy ? It appears that ...
0
votes
1answer
18 views

the analog to geodesic problem but with the area ?? for a variational problem

so we have that if we minimize the functional $$ S= \int_{a}^{b}\sqrt{g_{a,b}\dot x_{a}\dot x_{b}}$$ then the Euler Lagrange equations are $$ \frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu ...
2
votes
2answers
25 views

Exterior derivative of a coordinate function

I'm starting to learn about differential forms. From what I understand the coordinate differential forms $dx^1, \dots, dx^n$ are actually the exterior derivatives of the coordinate functions $x^1, ...
0
votes
0answers
15 views

Uniqueness of integral curve

Given a vector field $X$ on a smooth manifold $M$ and a point $p \in M$, we know that there exist an open neighborhood $U$ of p, an $\epsilon >0$ and a unique local flow $F : U \times (-\epsilon, ...
1
vote
0answers
45 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition ...
2
votes
0answers
44 views

Finding the equation of and drawing a cardiod

I have attempted drawing this many times and cannot come up with a cardioid. I believe the AM1 and AM2 are what is confusing me. I honestly have no idea where to start in order to find the equation. ...
2
votes
1answer
37 views

An analogue to the Koszul formula in the “wrong” degrees

Let $M$ be a smooth (closed, connected) manifold, $b\in\Omega^k(M)$, $P\in\Gamma(\Lambda^pTM)$ and $Q\in\Gamma(\Lambda^qTM)$ such that $p+q=k-1$. We denote by $[,]$ the Schouten-Nijenhuis bracket ...
1
vote
1answer
45 views

Differential of Sum of Two Functions is Sum of Differentials

Let $M$ be a smooth $n$-manifold and $f, g:M\to \mathbf R^n$ be smooth functions on $M$. Let $p$ be a point on $M$. I want to show that $d(f+g)_p=df_p+dg_p$ without passing to a chart about $p$. ...
1
vote
1answer
39 views

Straight line segment

I want to show that the straight line segment joining two points $p_1$ and $p_2$ in a plane is the shortest path between $p_1$ and $p_2$. I have tried the following: The straight line segment ...
2
votes
1answer
51 views

Relation between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?

Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the usual Lie algebra of vector fields on $M$; that is ...
1
vote
1answer
23 views

1-dim Vector Bundle sufficient condition to be trivial

I'm a physics student studying differential geometry. I'm trying to understand how vector bundles work, I have the following exercise. Let be $ L $ a $1$-dim vector bundle on $M$. Prove that if ...
2
votes
0answers
46 views

The limit of a uniform convergent sequence of isometries is an isometry (problem 6-3 of Lee's “Riemannian manifolds”)

I'm trying to prove the following theorem: let $f_n : M \to N $ a sequence of isometries of Riemannian manifolds that converges uniformly to a function $f:M \to N$: prove that $f$ is an isometry too. ...
2
votes
2answers
79 views

What is the space curve with curvature and torsion obeying

$ \kappa = \cos s, \tau = \sin s $ and passing through (1,0,0), TNB triad identity matrix? previous link When numerically computed it looks like a catenoid surface of revolution for all ...
3
votes
0answers
65 views

What is the formal definition of tangent hyperplanes?

My questions arise from 11.21 DEFINITON and 11.22 THEOREM ,both of them presented on p341 An Introduction To Analysis W.R.Wade 3ed. Question 1. Considering 11.21 DEFINITON, Let ...