1
vote
0answers
13 views

Topology of the intersection of toric arrangement

Hope someone will help me in the solution of the following question. I'm working on some topological problem involving the topology of the intersection of some characters of the torus. I want ot find ...
0
votes
0answers
25 views

Produce one smooth curve on one triangle mesh

I hope to get one smooth curve on one triangle mesh. I get one path on the mesh at first. The path consists of vertices of the mesh. I can see the path from the image below. Each one green dot ...
0
votes
1answer
37 views

When are two submanifolds “the same”?

Consider two smooth submanifolds $N\subseteq\mathbb R^n$ and $M\subseteq\mathbb R^m$. Let there be a function $\varphi\colon N \to M$ that is bijective. Which properties does the function $\varphi$ ...
1
vote
1answer
59 views

Distance between two points on the Clifford torus

How can I obtain the distance between two points $\mathbf{x}=(x_1,x_2,x_3,x_4)$ and $\mathbf{y}=(y_1,y_2,y_3,y_4)$ that belong to the $2$-torus $\mathbb{S}^1\times \mathbb{S}^1$? This is, I want to ...
3
votes
0answers
45 views

Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature ...
0
votes
1answer
14 views

Coordinate dependence of the volume of parallelotope

It is well known that for $n$ vectors $v_1, \ldots, v_n$ in $\mathbb R^n$, the determinant of the matrix $A = (v_1 \ldots v_n)$ [i.e. with the vectors as columns] is related to the volume of the ...
2
votes
1answer
26 views

How to derive the standard symplectic form on a 2-sphere in cylindrical polar coordinates?

I have already know that at a point $p\in S^2$ with $u,v$ the tangent vectors at $p$. Then the standard symplectic form is $\omega_p(u,v) :=\langle p,\,u\times v\rangle$ where $u\times v$ is the outer ...
2
votes
1answer
42 views

How to prove it's not a manifold?

In $R^{3}$ , let $Y_{r}$ be the set of points at distance $r>0$ from the circle $C= \{ \left(x,y,z\right) ; x^2+y^2=1,z=0 \}$ i.e. a doughnut which may be too fat. How to prove that when $r \geq ...
3
votes
1answer
52 views

Notion of a distribution as acting on tangent spaces

I'm reading a paper that uses distributions in a way I'm not entirely comfortable with. To be precise, I'm not sure what definitions the author is working with and can't find any natural way to fill ...
0
votes
0answers
27 views

“Circle” on pseudosphere

How should parametrization of the 2 parameter surface of a pseudosphere ("latitude" u and longitude v) change to result in a 1 parameter curve of constant geodesic curvature? EDIT: In other ...
0
votes
1answer
68 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
1
vote
1answer
34 views

Hausdorf Dimension of a manifold of dimension n?

Let's say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus ...
1
vote
1answer
57 views

Curvature proof of a convex plane curve

Having a little trouble with part b. Is there a way to show that this curve would be arc length paramaterized? I am assuming that we cannot say this. If it is not we can take alpha', alpha'' and ...
0
votes
2answers
30 views

Curve in union of hyperplanes

If a smooth curve $\gamma: [0,T] \to \mathbb R^n$ is contained in the union of hyperplanes $$ \bigcup_{i=1, \dots, N} H_i$$ does it then follow that one can always find time intervals $[t_0, t_1]$ ...
0
votes
0answers
24 views

On Steiner Surface

I would be very grateful if you help me with explicitly proving that the Steiner surface is a topological manifold. Thanks in advance!
1
vote
3answers
76 views

Examples of smooth curves of genus $0$ and degree $d>2$.

Can we provide a source of explicit examples ? The degree assumption $d>2$ means that I would like to see examples which are not conics.
2
votes
1answer
28 views

Finding an isometry that maps one circle to another.

I have a problem goes as follows: Consider the unit speed curve $$\boldsymbol{r}(s)=\left(\frac{4}{5}\cos(s),1-\sin(s),-\frac{3}{5}\cos(s)\right).$$ Find an isometry $f$ such that ...
0
votes
0answers
34 views

A question on relating $N$-Sphere with a $(N-1)$-cell in $\mathbb{R}^{N-1}$

Let there be a $N$-Sphere in $\mathbb{R}^N$. Every point in it is a unit vector in $\mathbb{R}^N$. Every real valued function $f$ defined on this sphere accepts a unit vector $\hat{a}\in\mathbb{R}^N$ ...
1
vote
0answers
51 views

geometrical consequences of nonpositive or negative Ricci curvature.

well,that's pretty much the question. I'd like to know if somebody could point me out if there's any geometrical implications following an upper bound on the Ricci curvature on a riemannian manifold. ...
0
votes
0answers
48 views

Arc length for a function $f:\mathbb{R}^2 \to \mathbb{R}^2.$

Assume $f:\mathbb{R}^2 \to \mathbb{R}^2$ is $C^1.$ Is there a formula for the length of the subset of $\mathbb{R}^2$ given by $$ \{f(x,y) \in \mathbb{R}^2:a_1\leq x\leq b_1, a_2\leq y \leq b_2\} ? $$ ...
1
vote
1answer
40 views

Variable Pitch Helices

Is it necessary for a helix to have constant pitch? If it is not so, what would be equation of a variable pitch helix?
0
votes
1answer
46 views

What is the hyperbolic plane equivalent to translation in euclidean space

in euclidean plane one can move polygons like rectangles, triangles etc. around by isometries, e.g. translations. For instance if we consider a rectangle with midpoint $0\in\mathbb{R}²$ then the image ...
0
votes
1answer
24 views

Compute the isotropy representation

Suppose $SU(1,1)$ acts on the open unit disc $\mathbb{D}$ in the natural way, by linear fractional transformations. The isotropy group is $U(1),$ since it stabilizes the point $0.$ I am trying to ...
0
votes
1answer
154 views

Compute the derivative of Plucker Embedding

Let V be an n-dimensional vector space over $\mathbb{R}$, and $$\Psi: G(k,V)\rightarrow \mathbb{P}(\Lambda^k V)$$ be the Plucker embedding, where $$L=span \{u_1, ..., u_k\} \mapsto \Psi(L)=[u_1 ...
3
votes
3answers
74 views

Point on an ellipsoid closest to line

The $2D$ case is not a problem: $$\ P(t) =(x,y)= s + t v = <s_x+tv_x, s_y+tv_y> $$ $$\ F(x,y) = (\frac{x}{a})^2 +(\frac{y}{b})^2 -1 = 0 $$ $$ \nabla F(x,y).v =0 $$ Finally solve for $y$ in ...
22
votes
2answers
460 views

Convex surface on which any two points $a,b$ can be joined by a curve of length $(\pi/2-\epsilon)|a-b|$

I am trying to solve an exercise on page 13 of the book Metric structures on Riemannian and non-Riemannian spaces by Gromov. Construct a closed, convex surface $X$ in $\mathbb R^3$ such that any ...
1
vote
1answer
31 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
1answer
32 views

Definition of a geodesic ball?

I think it goes along the lines of: a ball made of a series of flat sides. Also is a geodesic ball and geodesic dome the same thing?
0
votes
1answer
39 views

Roundness in Taxicab Geometry

I was just wondering whether circles are considered "round" still in taxicab geometry. I know that "roundness" is probably not a well-defined term and I know what a circle /appears/ to look like in ...
2
votes
2answers
34 views

Basic geometry proof about tetrahedron

Suppose tetrahedron ABCD such that AB is orthogonal to CD and AC is orgthongal to BD. Show that AD is orthogonal to BC. So i made a picture of a tetrahedron in 3 space and sort of look down at it ...
1
vote
1answer
79 views

Geometric interpretation of Laplace's formula for determinants

Coming from the geometric point of view, the determinant of an $n \times n$-Matrix computes the volume of an parallelepiped spanned by the columns of the matrix. In context of this question, let the ...
0
votes
1answer
27 views

convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
2
votes
0answers
42 views

properties of “plane curves” in the sphere $ S^2$

Just as there is a version of the theorem Jordan curve in $S^2$, will exist a version of the isoperimetric inequality? Would greatly appreciate any reference
0
votes
0answers
50 views

Antipodal map commutes with antipodal map? [duplicate]

Suppose we have a closed form $d\omega$ on $S^{n}$, and antipodal map $i: S^{n} \to S$ n i.e $i:x \to −x$. How to see that the external differential commutes with antipodal map?
1
vote
0answers
53 views

All differentiable functions on $\mathbb{S}^n$

Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...
0
votes
0answers
18 views

If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler > structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which acts on $M$ and preserves complex structure then If $M$ has Kaehler manifold, then the symplectic quotient of $M$, i.e, ...
1
vote
1answer
77 views

Reference about Gauss-Bonnet-Chern theorem.

I would like to get some references which explains Gauss-Bonnet-Chern theorem and its original proof by Chern. I tried to read his paper published in 1944 "A Simple Intrinsic Proof of the Gauss-Bonnet ...
1
vote
1answer
57 views

Differential geometry unit vector

Why is $$e_\mu=\partial_\mu$$always said to be the unit vector ? Doesn't the size of the vector $\partial_\mu$ kindoff depend on the underlying manifold ?
0
votes
0answers
25 views

the induced verctor field of the compose map

Let $M$ be a manifold, $\phi_t, \varphi_t: M\to M$, they induce two vector field, $$ \frac{d}{dt}\phi_t=X_t\circ \phi, \frac{d}{dt}\varphi_t=X_t\circ \varphi, $$ then what's the vector field induced ...
2
votes
0answers
104 views

Equation for the Maximum-Area Rotor in a Wankel Engine

I am attempting to construct a specialized Wankel Engine in Autodesk Inventor. For my particular project, the rotor must take up the maximum area, in order to separate the air spaces of the top and ...
0
votes
1answer
17 views

the zeros of a mean-value vanshing function

Let $F:[0,1]\times M\to \mathbb{R}$ be a smooth function, where $M$ is a smooth manifold, if we have $\int_{M}F(t,x)d\mu=0$ for all $t\in[0,1]$, where $\mu$ est a volume form on $M$, alors does there ...
0
votes
1answer
44 views

Kernel Of surrjective linear bundle map?

$E$ is a vector bundle on $M$. let $\phi:E\longrightarrow TM$ be a surjective linear bundle map. Is $\ker\phi$ vector sub-bundle of $E$?
1
vote
0answers
32 views

Motivation for tensor density

Wiki has provided the basic definitions of the tensor density, but what I really want to know is the motivation and the advantage of this concept. Could anyone give me some ideas?
0
votes
1answer
74 views

Why the geodesic curvature is invariant under isometric transformations?

As I know the geodesic curvature $$ \kappa_g = \sqrt{det~g} \begin{vmatrix} \frac{du^1}{ds} & \frac{d^2u^1}{ds^2} + \Gamma^1_{\alpha\beta} \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \\ ...
1
vote
0answers
44 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
2
votes
1answer
99 views

Prove the Gaussian curvature $K=0$

If two families of a geodesics on a surface intersect at a constant angle $\theta$, prove that the Gaussian curvature of the surface is zero, i.e. $K=0$. Please explain how to show the question. ...
0
votes
0answers
25 views

How could I calculate displacement along a 2D polyline by integrating each dimension separately?

This question's field of application is GPS trajectory analysis, but I'll try to give it a more abstract mathematical treatment. Suppose the trajectory of an object in 2D plane is described by a ...
0
votes
1answer
63 views

Regular Parametrization of a Sphere

Is there a function $f:U→ \mathbb{R^3}$, such that: (1) U is an open connected subset of $ \mathbb{R^2} $; (2) f is $ C^r , r≥1$; (3) the Jacobian of f is of maximal rank at all points of U; (4) ...
1
vote
0answers
68 views

principal axis of a volume from moments of inertia

I'm trying to calculate the expression to find the principal axis of a volume via its moments. In the 2D case I can formulate the problem by expressing the moments around arbitrary axes $x' = x \cos ...
2
votes
1answer
30 views

Given two closed curves, when is their minkowski sum differentiable?

Suppose you are given closed curves, $\gamma_1$ and $\gamma_2$, which define convex figures in the plane. If we take the minkowski sum of $\gamma_1$ and $\gamma_2$, when is the resulting curve ...