0
votes
0answers
2 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
41 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
31 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
30 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 ...
-1
votes
0answers
33 views

Finding the geographical coordinates

I have two circles $C_1$ and $C_2$ on the surface of the earth (sphere) intersecting at geographical coordinates $A$ and $B$ and also center of $C_1$ lies in $C_2$ and vice versa. I want to find the ...
0
votes
1answer
23 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
72 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
70 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 ...
17
votes
2answers
297 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
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
1answer
28 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
36 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
31 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
70 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
23 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
41 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
71 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
56 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
23 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
82 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
43 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
65 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
43 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
91 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
21 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
60 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
65 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 ...
0
votes
0answers
25 views

Global section $$s:T^*G\to Sp(T^*G)$$

Let $G$ be a Lie group and then how can we define the global section $$s:T^*G\to Sp(T^*G)$$ Where $Sp(T^*G)$ means symplectic frame bundle of $T^*G$.
0
votes
1answer
38 views

Showing the product rule for the Euclidean connection wrt the Euclidean metric.

I'm confused about a few things in Lee's book on Riemannian geometry. On page 67, Lee writes that it is easy to compute the following in terms of the standard basis where $\overline{\nabla}$ is the ...
1
vote
1answer
70 views

Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermition form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write ...
1
vote
0answers
42 views

Lateral area of a cone with oblique base (elliptical base)

Can someone help me find the lateral area of a straight circular cone with oblique base ( inclined elliptical base) as seen at the following link. ...
0
votes
0answers
56 views

compact surface of revolution

I have to prove that a compact surface of revolution is diffeomorphic to a sphere or to a torus. And show that $\int_{S} K dA= $ =$\{4\pi,$ if S is spherical type 0, if S is toric type $\}$, ...
1
vote
1answer
52 views

Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
0
votes
0answers
28 views

How to prove a parallel $(u=u_0) $it self curvature?

My name is Gita, and I had aproblem with my math. and need help, I know that a parallel $C$ in a surface of revolution in $M$ be a geodesic if and only if $f'(u_0)=0$. and $C$ is non arc lenght ...
1
vote
0answers
27 views

Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
1
vote
0answers
29 views

Show an immersion is locally one to one using the inverse function theorem

Using the inverse function theorem, show that an immersion is locally one to one. I am really struggling with this homework question can anyone give me a hint?
2
votes
1answer
56 views

The relation between principal curvature and curvature tensor?

To me, there are two systems of curvature of a surface, one is consist of 'principal curvature, mean curvature, Guass curvature, normal curvature' while the other is consist of 'curvature tensor'. I ...
1
vote
1answer
95 views

Prove that the curves of the family $v^3/u^2=k$ are geodesics on a surface

Prove that the curves of the family $v^3/u^2=k$ where $k$ is a constant are geodesics on a surface with the metric $$v^2 \, du^2-2uv \, du+2u^2 \, dv^2$$ where $u,v \gt 0$.
1
vote
0answers
41 views

Is $\nabla_{X}:\Gamma(E) \to \Gamma(E)$ continuos aplication?

Be $\Gamma(E)$ smooth sections space of an vector bundle, $\Gamma(E)$ with Fréchet topology. Gived a conexion in E and a smooth vector field $X$, Is $\nabla_{X}: \Gamma(E) \to \Gamma(E)$ continuos ...
10
votes
2answers
133 views

Analogue of a parabola on a sphere?

Parabola: the set of points in the plane that are equidistant from a line, called the directrix, and a point, called the focal point, not on the line. Suppose we try to replicate this on a sphere: ...
3
votes
1answer
102 views

Showing to be Unit-sphere

First and second fundamental forms are both $du^2 +\cos^2 u dv^2$ I want to show that the surface is a part of the unit sphere. What I did is following; $E=L=1$ $F=M=0$ $N=G=\cos^2 u$ ...
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
60 views

Tangent space of the tangent bundle

Let $M$ be a differential $k-$manifold in $\mathbb R^n$, $g:TM\rightarrow\mathbb R$ given by $g(x,v)=|v|^2$, where $|\cdot|$ is the usual norm in $\mathbb R^n$. Show that $Reg(g)=Reg(g_{\partial ...
0
votes
2answers
88 views

why it is a projective space?

The unite sphere bundle of $TS^2$ , the tangent bundle of the 2-sphere, is the real projective space, $RP^3$. I can not understand the reason of it. And why the complement of the unit disc bundle of ...