# Tagged Questions

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### 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. ...
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### 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 $$\{f(x,y) \in \mathbb{R}^2:a_1\leq x\leq b_1, a_2\leq y \leq b_2\} ?$$ ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $T^*G$.
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 $\}$, ...
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### 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 $k_2$, ...
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### 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 ...
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### 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 $\mathbb{S}^n_+$, ...
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### 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?
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### 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 ...
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### 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 $u,v \gt 0$.
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### 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 ...
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### 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: ...
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### 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; $E=L=1$ $F=M=0$ $N=G=\cos^2 u$ ...
### 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 ...