1
vote
1answer
25 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
35 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
25 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
62 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
20 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
39 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
1
vote
0answers
54 views

Special expression for 3-linear symmetric map $T(X,Y,Z) = \langle -Jh(X,Y),Z \rangle$ [Ejiri]

For a project in Riemannian Geometry, I have been working out the details of a paper by Ejiri (http://www.ams.org/journals/proc/1982-084-02/S0002-9939-1982-0637177-8/S0002-9939-1982-0637177-8.pdf) ...
0
votes
0answers
49 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
67 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
53 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
73 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
39 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
59 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
42 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
90 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
20 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
61 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
29 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
63 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
31 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
20 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
28 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
53 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
40 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
130 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
31 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
54 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
87 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 ...
4
votes
1answer
99 views

Geodesic eqautions and length of a curve in geodesic coordinate system.

About geodesic coordinates: Let S be regular surface. $p\in S$ $\gamma$ be unit speed geodesic on $S$ with parameter $v$ and $\gamma (0)=p$ $\tilde \gamma^v$ be unit speed geodesic s.t. ...
3
votes
0answers
146 views

How can higher-dimensional projection maps be described mathematically?

New question: (resulting from discussions with Sabyasachi) I am wonder how can higher-dimensional projection maps, analogous to for example the Mercator, Miller, Behrmann projections, can be ...
5
votes
1answer
66 views

Problem about parallel curve- differential geometry

Let $\alpha (s)$ , $s\in [0,L]$, be a smooth positively oriented regular Jodan curve which is arc-length parametrized. The curve $\beta(s)=\alpha (s) +\lambda n(s)$, where $\lambda$ is a positive ...
21
votes
0answers
608 views

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
1
vote
2answers
84 views

Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why?

What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this?
2
votes
1answer
38 views

Tangent plane and normal

I need to find the tangent plane and a unit normal vector to $ r(u,v)=((2+\cos(u))\cos(v),(2+\cos(u))\sin(v),\sin(u))$ at $(2+\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}) $. $\frac{\partial r}{\partial ...
3
votes
1answer
54 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
3
votes
2answers
83 views

Find the maximum angle possible

$P$ is a point on the $Y-axis$ . Find the maximum possible value of $\angle APB$ where $A=(1,0)$ and $B=(3,0)$. Here is how I solved the problem. Suppose $P=(0,k)$ . Then using the cosine formula we ...
0
votes
0answers
28 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 ...