0
votes
0answers
28 views

clear statement about the relation between curvature and rotating vectors along a loop

in my research, I need to understand the relation between the formal definition of Riemann curvature tensor ($R_{jkl}^i$)and the ``vector rotation" approach: parallel transport a vector along a loop, ...
8
votes
1answer
108 views

Why are Banach manifolds not so popular?

Why are Banach and Frechet manifolds studied not even remotely as much as Euclidean manifolds? I assume like many other mathematical subjects, theory of manifolds has been developed much more than the ...
3
votes
3answers
59 views

$T^2\times S^n$ is parallelizable

This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that $T^2\times S^n$ is parallelizable for all $n\geq 1$? Is there a way to find $n+2$ linearly independent vector ...
1
vote
1answer
29 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
1
vote
0answers
41 views

Tangent bundle is orientable

I am having some trouble finishing a proof that the tangent bundle of any manifold is orientable. What I've done so far is calculate the transition function between two standard charts on the bundle. ...
2
votes
1answer
53 views

Graph theoretic view on manifold triangulations

To make the question (hopefully) clearer, I reformulated it: Some triangulation $T$ of a smooth manifold $M$ is a piecewise linear manifold, because smooth manifolds are topological manifolds. Such a ...
1
vote
1answer
29 views

Examples of finding norm-minimizing surfaces and Thurston polytope

Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is ...
1
vote
3answers
58 views

What is a “control point”?

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ...
3
votes
0answers
95 views

measurable function induces a measurable bundle

Greetings I am preparing a work on bundles and I found this statement Let $V$ a topological vector space with $\dim V=n$ and $(E,\pi,M)$ a vector bundle continuous over $M$ (compact space). If ...
4
votes
2answers
136 views

Why do folded concentric circles and rectangles form a hyperbolic paraboloid?

Here is a "self-forming" origami that I made from folding concentric circles - it would also happen if I folded concentric rectangles. How can the fold shapes such a saddle-like geometry?
0
votes
1answer
30 views

Is there a smooth map from the square to the deltoid?

Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 ...
2
votes
1answer
82 views

Calculus on manifolds

I am studying manifold theory ( smooth manifold). As I study the subject, I feel more and more confused. I am not yet at chapter dealing with deferential forms and integration. I can understand that ...
0
votes
0answers
32 views

Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
0
votes
0answers
29 views

A question about complex polarization

Let $M$ be a smooth manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P \cap ...
1
vote
0answers
42 views

What are different geometric interpretations for the sample variance?

I primarily work with non Euclidean data and am looking to extend concepts of 'variance' to Riemannian manifolds. I am aware of Karcher variance, but I need efficient ways to solve for it. For ...
0
votes
1answer
39 views

Curve in $\mathbb{P}^{n}(\mathbb{R})$, differentiable manifolds

I need a book which contains the demonstration that if $\Gamma$ is a curve in $\mathbb{P}^{n}(\mathbb{R})$ defined as $F(x_{1}, \cdots \ \ , x_{n+1}) = 0 $ , with $F$ homogeneous polynomial, then ...
0
votes
0answers
29 views

How to find a curve inside a non-convex

I want to connect two points in a space within the space. If the space is convex, I can simple draw a line between them. But how about a non-convex space. How can I find a curve connecting these two ...
3
votes
1answer
117 views

Partial derivative of a function on manifold

Bishop and Goldberg define ("Tensor analysis on manifolds") the partial derivative of a smooth function on a manifold $M$ in the following way: $\partial_i f= \frac{\partial ...
2
votes
1answer
201 views

Tangent space of Grassmannian $Gr_k (\mathbb{R}^n)$

I am trying to show that the tangent space of the Grassmannian $Gr_k (\mathbb{R}^n)$ at $L,$ is naturally/canonically isomorphic to $Hom(L,\mathbb{R}^n/L).$ However, I cannot see even intuitively what ...
0
votes
1answer
39 views

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature.

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature. I dont have enough idea. Please explain the question clearly. ...
4
votes
2answers
162 views

Most important aspects of differential geometry for general relativity

I'm an undergraduate getting ready to take a graduate course in general relativity next quarter. I purchased Wald's General Relativity (who incidentally will be teaching the class) in order to get a ...
-1
votes
1answer
150 views

Compute the geodesic curvature of any sphere on a sphere.

Compute the geodesic curvature of any sphere on a sphere. Again there exists its answer, but not understandable for me. Please explain it explicitly. Thank you so much. (If required, i can post ...
1
vote
1answer
73 views

Second fundamental form question.

Honestly, I dont have any idea for that question I posted. Please can someone help me solving the question. Thank you.
0
votes
1answer
103 views

Transition map for Möbius band in differential geometry.

Calculate the transition map $\phi$ between the two surface patches for the möbius band. These two surface patches are the following $U=\{(t,\theta) \ | -1/2\lt t\lt 1/2,\ \ 0\lt \theta \lt ...
3
votes
1answer
49 views

write the trasition map $\phi$ between $\sigma_1$ and $\sigma_2$. Verify $\det( J(\phi))$ and find $T_p(S)$.

Sphere $$S=\{(x,y,z) \mid x^2+y^2+z^2=R^2\}$$ $$ \sigma_1(u,v)=(u,v, \sqrt{1-u^2-v^2}) \\ \sigma_2(u,v)=(\tilde u, \sqrt{1-\tilde{u}^2 -\tilde{v}^2}, \tilde v) $$ I guess $\{\sigma_1, \sigma_2\}$ ...
2
votes
1answer
212 views

Tensors constructed out of metric other than the Riemann curvature tensor

Let $(M,g_{ab})$ be a Riemannian (or Pseudo-Riemannian) manifold and let us define a tensor field as 'something' that transforms in an appropriate way under coordinate transformations. (This is how ...
-1
votes
1answer
136 views

The double cone is not a surface.

My question is that A double cone ( also named as "circular cone") is not a surface. I know its reason. But I cannot show this mathematically. Suppose $\sigma : U \to S\cap W$ Is a surface ...
0
votes
1answer
42 views

Show that a paraboloid is asurface .

That I know about paraboloid is all in the picture. I wrote its surface patch. (Hopefully, it is correct) From there, what do I need to do in order show that a paraboloid is a surface. ...
1
vote
1answer
47 views

Taylor expansions on manifolds..

can one consider Taylor expansions of functions defined between smooth manifolds? If so, is there a reference for learning more about it? Thanks
3
votes
2answers
121 views

How does a smooth structure on a subset of a manifold determine its status as an immersed submanifold?

As titles are limited to 150 characters, allow me to rephrase my question in a way that is hopefully more precise: Given a $d$-dimensional smooth manifold $M$ and some $k$-dimensional subset ...
2
votes
2answers
36 views

Can I identify $S_k(V)$ with an homogeneous space?

I'm in trouble with a question: Let $V$ be an $n$-dimensional vector space over $\mathbb R$. Can I identify the manifold, $$S_k(V):=\{(X_1, \ldots, X_k): X_1, \ldots, X_k\in V\ \textrm{are linearly ...
0
votes
0answers
15 views

How can I see the set $S_k(V):=\{A: \mathbb R^k\rightarrow V: A\ \textrm{is linear and}\ \textrm{ker}(A)=\{0\}\}$ as an homogeneous space..

how can I see the set $S_k(V):=\{A: \mathbb R^k\rightarrow V: A\ \textrm{is linear and}\ \textrm{ker}(A)=\{0\}\}$ as an homogeneous space? Thanks..
0
votes
1answer
94 views

$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$

Show that for a curve lying on a sphere of radius r with nowhere vanishing torsion, the following equation is satisfied: $$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$$ Please ...
3
votes
1answer
55 views

Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. Show that, if $\gamma$ is a unit-speed plane curve, ...
0
votes
0answers
26 views

Manifolds : Show that a maximal $C^r$-atlas containing a $C^r$-atlas $A$ is unique.

So I want to show that a maximal $C^r$-atlas containing a $C^r$-atlas $A$ is unique. I believe I have shown this, but in my proof I require the fact that the charts of $A$ cover our manifold in ...
1
vote
1answer
87 views

How to show that the limaçon has only two vertices.

Question: Show that the limaçon has only two vertices. I researched what is limaçon. And I reached the following result; Note that I only know that The limaçon is the parametrized curve ...
2
votes
1answer
106 views

Writing a parametrization of the cissoid by using $\theta$

The cissoid of Diocles is the curve whose equation in terms of polar coordinates $(r,\theta)$ is $$r = \sin\theta \tan\theta, −\pi/2 < \theta < \theta/2$$ Write down a parametrization of the ...
0
votes
0answers
62 views

French translation, and what is the curvature of a metric?

I have a french paper to read. There is the notion of une collection des courbures des métriques $g_t$. Now I would guess that this refers to a collection of curvatures of metrics $g_t$, however ...
1
vote
0answers
37 views

Does there exists a simple imbedding theorem for general topological $n$-manifolds?

I am interested in finding some paper or book where i can find how to build an imbedding $e\colon M^n \hookrightarrow \mathbb{R}^q$ of an arbitrary Hausdorff topological $n$-dimensional manifold $M$ ...
2
votes
2answers
72 views

Problem with normal bundle to sphere $\mathbb S^n\subset \mathbb R^{n+1}$

I'm in trouble for understanding the normal bundle to $\mathbb S^n\subset\mathbb R^{n+1}$. By definition $$\nu(\mathbb S^n)=\bigcup_{p\in\mathbb S^n}\nu_p(\mathbb S^n),$$ where $\nu_p(\mathbb ...
2
votes
1answer
130 views

Construct a map from unit disk to upper half-plane

I want to construct this map in high-dimensional case. Let $D=\{x \in \mathbb{R}^n:|x|^2<1\}$,and $H=\{u\in\mathbb{R}^n:u^n>0\}$. Well, it is quite clear when $n=2$, but I find it is hard for me ...
4
votes
1answer
103 views

Is there a name for this particular class of topological space?

This is a simple question, but I can't figure out the name for this class of topological space. Say you start with the affine space $\mathbb{R}^n$ for finite n, and equip it with a metric. Now, say ...
0
votes
0answers
27 views

extremal points on a manifold intrinsiclly

I am wondering if there is a geometric object for real analytic manifolds that characterizes extremal points of the manifold intrinsically. For instance, suppose I live in the manifold, can I ...
2
votes
1answer
212 views

Definition clarification on orientation on a manifold.

I have been trying to self-learn differential geometry. I think I may have misunderstood/missed out on something along the way. It is said that for $X$ an $n$-form, $M$ a differentiable manifold, ...
4
votes
2answers
66 views

How to use Mayer-Vietoris to show $\chi(X)=2\chi(M)-\chi(\partial M)$ where $X$ is the double of $M$?

I'm in trouble with the following problem: Let $M$ be a manifold with compact boundary $N$ and let $X$ be the double of $M$, that is, the manifold without boundary one gets by glueing $M$ with ...
0
votes
0answers
46 views

isomorphism of two compex line bundles

I am looking for some non-trivial examples of Line Bundles and an example about isomorphism of two line bundles. With details
2
votes
1answer
51 views

Exact sequence arising from symplectic manifold

Let $M$ be a symplectic manifold, why folloing sequence is exact? $0\to \mathbb{R} \to C^\infty (M)\to A\to 0$ Which $A$ here is the set of Global Hamiltonian vector fields.
1
vote
1answer
61 views

integrability of ker $\omega$ in symplectic case

How can we prove that if $(M,\omega)$, is pre-symplectic and d$\omega=0$ then ker$\omega$ is integrable?.
3
votes
0answers
112 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ ...
0
votes
1answer
59 views

Imbeddings of $n$-dimensional topological manifolds in $(2n + 1)$-euclidean space

H. Whitney proved that any $n$-dimensional smooth manifold $N$ can be imbedded in $(2n + 1)$-euclidean space (without any compactness assumption). If we consider the case of topological $n$-manifolds ...