Tagged Questions

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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4answers
36 views

Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
1
vote
0answers
18 views

Is (0,0) of $V(x-y^2)$ a smooth point?

I'm pretty sure it is a smooth point since given $f(x,y)=x-y^2$ the gradient $df=(1,-2y)$ is always non-singular. I'm asking because page 22 of Principle of Algebraic Geometry says: ...
4
votes
1answer
30 views

Weak tangent but not a strong tangent

Question: Show that $\alpha(t)=(t^3,t^2)$, $t\in \Bbb R$, has a weak tangent but not a strong tangent at $t=0$. Definitions from this answer: (Weak tangent) $\alpha: I \to \Bbb R^3$ has a weak ...
0
votes
1answer
16 views

Restricting the DeRham cohomology class of a submanifold to a coordinate neighborhood.

Suppose $M$ is an $n$-manifold and $A$ a $k$-dimensional submanifold, both compact and oriented. Let the deRham cohomology class of $A$ be denoted $[\phi_A]$. The class is defined by ...
3
votes
2answers
47 views

Definition of a Manifold from Guillimen Pollack

I have been studying differential topology from Guillimen and Pollack (GP). Unlike many other books that define differentiable manifolds using maximal atlases GP starts by saying $ X \subset R^{N}$ ...
3
votes
1answer
122 views

Confusing Analysis proof

I have a question about a proof of the Beltrami-Enneper theorem: In the following $\nu$ is the surface-normal and $e_1,e_2,e_3$ the Frenet 3-frame. It states: Every asymptotic curve $c: I \rightarrow ...
1
vote
0answers
12 views

Parameterization of surface of revolution with constant mean curvature

Let $x(u,v) = (g(u), h(u) \cos v, h(u) \sin v)$ be a parameterization of a surface of revolution $M$, arising from rotating the regular curve $\alpha(u) = (g(u),h(u),0)$ around the $x$-axis with ...
0
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0answers
35 views

How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
1
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1answer
24 views

How to define the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$?

I read a paper and on page 9, the paragraph before Proposition 5, it is said that let the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$. How to ...
0
votes
1answer
47 views

Differentiating with respect to a vector

Hello i'm new to this forum and this is my first post. I was going over the transport theorems in fluid mechanics and there is one way in which you can convert reynolds transport theorem into a single ...
2
votes
2answers
42 views

Showing that a specific curve is regular.

Define the curve by $c(t):=(sin(pt)+r)(cos(qt),sin(qt))$ for $p,q \in \mathbb{Q}$ and $r\in \mathbb{R}$. Determine for which $p,q$ is the curve regular, i.e. $c'(t) \neq (0,0)$ for any $t\in ...
2
votes
1answer
37 views

Equivalent definitions of Euler characteristic for closed manifolds

It is well-known that the Euler characteristic of a closed manifold $M^n$, which can be defined as $\chi(M)=\sum_{k=0}^n (-1)^k \operatorname{dim}H^k(M)$, equals the intersection number ...
-1
votes
0answers
16 views

Translated stochastic process

Let $M$ be a (compact) Riemannian manifold and let $L$ be some second-order elliptic operator on $M$. Now for a vector field $v$, I can consider the flow $\Psi_t$ of $v$ and consider the following ...
1
vote
0answers
23 views

Total variation of real valued functions on a manifold

We can define the total variation of a function $f:\mathbb{R}^n\to \mathbb{R}$ as in Evans and Garriepy's "Measure Theory and Fine Properties of Functions" or in this Wikipedia article ...
1
vote
0answers
38 views

three vector fields with zero Lie bracket

Suppose I have three unit vector fields $u, v, w$ on a two-dimensional surface in $\mathbb{R}^3$. Are there simple compatibility conditions relating $v$ and $w$ such that there exists scalar fields ...
0
votes
1answer
41 views

Gaussian Curvature

I am able to show that if a curve lies in a plane then it's curvature at a point $p$ is $$\kappa=\lim_{\mu\to 0}\frac{\sigma}{\mu}$$ where $\mu$ is the length of a segment of the curve containing p, ...
0
votes
1answer
53 views

Minimal surface and Weierstraß parametrization

If I have $f(z) = 1$ and $g(z) = \frac{1}{z}$ and I am looking for a minimal surface on $\mathbb{C} \backslash \{0\}$ using the Weierstraß-Enneper representation of minimal surfaces. Now I was ...
2
votes
1answer
28 views

Euler characteristic of closed surface

Assume that you have a closed surface that can be covered by finitely many triangles. Then $K(p)= 6-val(P)$ where P is a vertex and $val(P)$ the number of edges that lead to this vertex. Now, I am ...
3
votes
2answers
65 views

How long is the curve that a creature walks?

I have a problem in solving mathematical problem. Take a ball with radius 60 cm. A creature walk from the southpole to northpole by following the spiral curve that goes once around the ball every ...
2
votes
1answer
41 views

Submanifold associated to blow up.

I 'm trying to understand the classical blow up given by $$X=\{(x,[y])\in \mathbb{R}^n \times \mathbb{P}_N / \hspace{0.2cm} \exists \lambda \in \mathbb{R} \hspace{0.3cm} \text{such that} ...
2
votes
0answers
33 views
+50

General form of a connection with zero curvature

I am looking for proofs of the following two theorems: Theorem 1. On a connected and simply-connected open set $\Omega\subset\mathbb{R}^3$, functions $L^p_{ij}\in C^1(\Omega)$ are given that satisfy ...
2
votes
3answers
54 views

normal bundle on a submanifold

Can you give me an example of a nontrivial normal bundle of a submanifold (of any manifold)? There is standard example of the core circle of a mobius band, but can you give an example of a submanifold ...
1
vote
1answer
35 views

On a flat surface, can a holonomy can be nontrivial around certain curves

On a flat surface, can a holonomy can be nontrivial around certain curves? How is this possible?
3
votes
0answers
28 views

Turning number VS winding number

To avoid confusion, here are the definitions of the objects in this question: 1) Let $\gamma:S^1\to\mathbb{R}^2\setminus\{0\}$ a smooth loop. The winding number of $\gamma$ is the number of times ...
0
votes
1answer
37 views

Applying Brower's Theorem for Invariance of Domain

I'm working through Manifolds and Differential Geometry by Jeffrey M. Lee. In a topological manifold, we have that every point is in an open set which is homeomorphic to $\mathbb{R}^n$ for some $n$. ...
7
votes
3answers
137 views

Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?

Is there any answer of this question around basic theory of differentiable manifolds?
2
votes
0answers
19 views

“Projection of metric” vs. “projection of curvatures”

Suppose we have a submanifold $M^n$ which is embedded in manifold $M^{n+2}$ and $g_{\mu \nu}$ denotes the metric of $M^{n+2}$. We know that the induced metric on the submanifold is defined by ...
1
vote
0answers
20 views

Topology of operator bundle?

I am trying to understand the family version of the Atiyah-Singer Index Theorem as described in the book "Spin Geometry" by Lawson/Michelsohn. In Part III.§8, they define the operator bundle $$ ...
0
votes
1answer
32 views

Hypersurface is not curved, normal vector

Let $S$ be a hypersurface in $\mathbb{R}^n$. Is there a simple way to say that $S$ is flat by describing the normal vectors on $S$? Like $S$ is flat if the normal vectors on $S$ are all identical..
9
votes
1answer
77 views
+50

Are compact complete geodesics closed?

Let $(M,g)$ be a compact Riemannian manifold. Is there an example of a geodesic $c:\mathbb{R}\to M$ s.t. $c(\mathbb{R})$ is compact, $c$ is NOT periodic (i.e. be NOT a closed geodesic) ?
1
vote
1answer
24 views

A question about the Integral geometry and geometric probability.

In the book: Integral Geometry and Geometric Probability, (p16-17), the author proved that the measure of randomly throwing three points P1, P2, and P3 on the plane such that the circumdisk and the ...
0
votes
0answers
15 views

alternate definition of winding number?

If c is a singular $1$-cube in $R^2-\{0\}$ with $c(0)=c(1)$ , show that there is an integer $n$ such that $c-c_{1,n}=\partial c^2$ for some $2$-chain $c^2$. Here $c_{R,n}=(R\cos 2\pi nt,R \sin ...
3
votes
1answer
29 views

The action of a Lie algebra on a manifold is a Lie algebra homomorphism. How to show it?

By definition, the action of a Lie algebra $\mathfrak g$ on a manifold $M$ is a Lie algebra homomorphism, $\mathcal A: \mathfrak g\rightarrow\mathfrak X(M), \xi\mapsto\xi_M$ such that the action map ...
0
votes
0answers
18 views

Implicit partial second derivatives from coupled equations

I have three functions defined in two variables. $f(x,y)$ $g(x,y)$ $h(x,y)$ I wish to find the partial derivatives $f_{gg}$, $f_{hh}$, and $f_{gh}$ and evaluate them at a particular point. In this ...
0
votes
0answers
21 views

Useful Coordinate Families on Lie Groups

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. We all know, since $\exp$ is a diffeomorphism in some neighborhood $V$ of $0\in\mathfrak{g}$, that we can cover $G$ in coordinate charts ...
0
votes
1answer
35 views

understanding the meaning of formal linear combination and tensor product

I have a question about understanding the meaning of formal linear combination. Let S be a set, the free vector space $\mathbb{R}\langle s\rangle$ on S is defined as the set of all formal linear ...
2
votes
1answer
27 views

Very basic question about submanifolds.

I'm beginning to study differential geometry and I'm a litle confused about the concept of submanifold of a differentiable manifolds. Can someone provide me an example of how to show that a non-open ...
3
votes
1answer
83 views

How useful is the Weierstrass representation of minimal surfaces?

Weierstrass representation of minimal surfaces says that if I have a holomorphic function $f: U \rightarrow \mathbb{C}$ and a meromorphic function $g: U \rightarrow \mathbb{C}$ such that $f g^2$ is ...
0
votes
0answers
20 views

Compact hypersurface in $\mathbb{R}^n$

Let $S$ be an $(n-1)$ dimensional hypersurface in $\mathbb{R}^n$. If we say that $S$ is compact, does this necessarily mean that $S$ has no boundary? Eg. $S$ can be a sphere but not a sphere cut in ...
0
votes
0answers
48 views

Generalized Stokes theorem applied to Tensor Moments

I am working with geometry and need to calculate 0th, 1st and 2nd moments in polyhedra, its polygons and its lines. From a previous answer in this forum, I understand that p-moments are: $ M^p = ...
3
votes
0answers
33 views

explicit (holomorphic) map revealing blow-up as a connected sum with $\overline{\mathbb{CP}}^n$

I am trying to understand the statement that a blow-up of a complex manifold $M$ at a point $p$ is equivalent to the connected sum of $M$ with $\overline{\mathbb{CP^n}}$ and, being a physicist, I need ...
-1
votes
0answers
34 views

Harmonic $k$-forms on torus

Find all harmonic $k$-froms on $n$-dimensional torus equipped with metrics $ds^2 = x_1^2 + \dots + x_n^2$. Harmonic means that Laplace-Beltrami operator $D = dd^* + d^*d = 0$.
1
vote
0answers
30 views

How to compute the classic differential of a section

Given a section $s:M\to E$ of a vector bundle, I'm trying to compute its differential, $Ts:TM\to TE$ in a local trivialization. To do this, I can write $$s=f^ie_i$$ For some local frame $(e_i)$ of ...
0
votes
0answers
39 views

Total curvature of a piecewise smooth planar curve

I am trying to show that the total curvature of a piecewise smooth planar curve $\alpha$ has the property that $\displaystyle \int_{C}\kappa $ $\text{d}s + \displaystyle \sum_{j=1}^{l} \epsilon_{j} ...
3
votes
1answer
33 views

Formal adjoint of divergence

We define the so-called conformal Killing operator $K$ mapping (1,0) vectors to (0,2) tensors by $$K(X)_{ab} = \frac{1}{2}\nabla_aX_b+ \nabla_bX_a -\frac{2}{3}(\text{div}X) g_{ab}.$$ Here $g$ is the ...
1
vote
0answers
26 views

Embedding counterexample

Lee writes on page 156 of Introduction to Smooth Manifolds: A smooth embedding is a map that is both a topological embedding and an immersion, not just a topological embedding that happens to be ...
1
vote
0answers
24 views

Calculating Integral Submanifolds

I have the vector fields $v_{1} = x \partial_y - y \partial_x + z \partial_w - w \partial_z$ and $v_{2} = z \partial_x - x \partial_z + w \partial_y - y \partial_w$ on $S^{3} \subset \mathbb{R}^4$. I ...
0
votes
2answers
37 views

How to orient a manifold in the Euclidean space?

I learned that the orientation of a smooth manifold is a smooth choice of an orientation for bases of tangent space. Also I sometimes read that an embedded manifold in $\mathbb{R^3}$ inherites an ...
1
vote
0answers
17 views

Continuity of length under uniform convergence

Let $(\phi_n):[0,1]\to \mathbb{R}^d$ be a sequence of injective curves, parameterized by constant speed, uniformly converging to a limit curve $\phi:[0,1]\to \mathbb{R}^d$. Assume: $\sup_n ...
2
votes
1answer
57 views

Vector fields on manifolds

I have recently started a course on differential geometry (from a physicists perspective) and I am having trouble convincing myself why vector fields are represented as differential operators on ...