Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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$h$-principle for isometric embeddings

All the references I have seen so far list the Nash $C^1$-embedding theorem as an example where the $h$-principle holds. The $h$-principle for a differential relation holds by definition, when the ...
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33 views

Example of sheaf hom not commuting with stalk

I came up with the following example and I wanted to know if it is correct. Let $\mathcal F$ and $\mathcal G$ be sheaves of smooth functions on $\mathbb R$. Consider the smooth function $f$ ...
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Lie group structure of the spin group

let $Cl_n:=T(\mathbb{R}^n)/I$ be the clifford algebra of $\mathbb{R}^n$ with the standard inner product. (Here $T(\mathbb{R}^n)$ denotes the tensor algebra of $\mathbb{R}^n$ and $I$ is the ideal ...
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30 views

$S^1$-curves on a Lie group $G$ under additive and multiplicative notation.

I have been trying to do computations for objects of the based loop group and have been embarrassingly frustrated by the following: Let $G$ be a compact, connected, simply connected Lie group with ...
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94 views

Spivak vol. 2 — expression of Riemann's quadratic function

I would very much appreciate it if someone could explain or at least indicate a proof of the following assertion in Spivak's ''Compr. Intro. to Diff. Geom.'', vol. 2, p. 171 (3rd ed., 2nd printing): ...
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220 views

Ambrose Singer Theorem

I wish to learn about holonomy groups of Riemannian manifolds and the Ambrose- Singer theorem. Please advise some references other than the original paper of Ambrose and Singer.
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27 views

Determine the direction of given parametrization.

I saw an example, which I posted below. First of all, I understand how to show paramtrized curve but I dont understand how to determine the direction of the parametrization. For example, how can ...
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29 views

Higher Order Torsion

Define an k-Torsion as a measure of how much a parametrically defined curve $x(t)$ where $t$ is a real scalar and $x$ is a vector in $R^n$ deviates from the locally encapsulating k-dimensional ...
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20 views

Functions from $[E_8]$ to $E_8$

Let $f: [E_8] \to [E_8]$ be a function between 4-manifolds with intersection form $E_8$. What we know (due to Rocklin) is that $[E_8]$ can't have any smooth structure. Questions: Is it true for all ...
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134 views

Surface orientation

Let $S_{1}$ and $S_{2}$ be two oriented surfaces ($N_{1}$ and $N_{2}$ their normal fields, respectively). We say that a local diffeomorphism $f$ : $S_{1}$$\rightarrow$$S_{2}$ preserves orientation if ...
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56 views

Why is $f:\mathbb{R}\to S^1$ $f(t)=(\cos(t),\sin(t))$ a local diffeomorphism?

An example in my book says that $f:\mathbb{R}\to S^1$ defined by $f(t)=(\cos(t),\sin(t))$ is a local but not global diffeomorphism. By the inverse function theorem, $f$ is a local diffeomorphism if ...
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42 views

Explanation required of the following definition:

This is a definition I encountered in a paper. I hope someone will be able to help me understand it. The authors assume a Frenet curve $\alpha(s)$ on a 3-D Riemannian Manifold as any non-geodesic unit ...
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117 views

derivative along a curve with respect to a given vector field

This is question 7 of Do carmo's differenital geometry of curves of surfaces 3.4. Define the derivative $w(f)$ of a differentiable function $f:U \subset S\rightarrow R$ relative to a vector field $w$ ...
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52 views

Prove equivalence of two conditions to be a smooth $k$-manifold $M^k \subseteq \mathbb{R}^n$

For the first couple classes of differential geometry, we have used the more concrete characterization of a manifold (given in #1 below). I am trying to prove that the following two conditions are ...
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49 views

A question on the idea of curvature of Riemann

In the book Mathematical Masterpieces, chapter 3, section 1, the authors have talked about the curvature and the ideas around it. They wrote If the curvature is given in $\dfrac{1}{2}n(n-1)$ ...
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62 views

Christoffel Symbol Not Disappearing

If I am given a vector field $\vec{A}(x,y) = x^2 \hat{e}_1 + y^2 \hat{e}_2 = (A^x,A^y)$, I'd like to calculate it's covariant derivative in the $r$ direction after expressing the vector field in polar ...
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181 views

Intrinsic and extrinsic properties of sets

Can a distinction between intrinsic and extrinsic properties of general sets a) be defined rigorously and b) be used fruitfully? (References?) An intrinsic property of a set $M$ is supposed to be ...
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283 views

$N$ equally spaced points on an ellipsoid

I would like to find a algorithm for determining the $(x,y,z)$ co-ordinates for evenly distributed $N$ points on the surface of an ellipsoid. These points must be spaced from its nearest neighbour ...
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53 views

Exponential intertwining of linear and local actions.

I am reading Duistermaat's 1973 paper on relating the convexity of the image of a moment map to the image of the fixed points of an antisymplectic involution. In that paper, the following comment is ...
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71 views

On the canonical neighborhoods

Suppose $M$ is a 3-dimensional manifold, John W. Morgan and Frederick Tsz-Ho Fong in their "Ricci Flow and Geometrization of 3-Manifolds" book as a definition of canonical neighborhoods have ...
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58 views

Immersion from $R^{2}$ to $R^{4}$

If we have an immersion from $R^{2}$ to $R^{4}$ defined by \begin{align} \notag f:(x,y) \to (x,y,x,y). \end{align} If basis of $R^{2}$ is $\{e_{1},e_{2}\}$ and basis of $R^{4}$ is ...
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662 views

Differential forms and wedge product and exterior derivative

Could anyone help me with some easy examples of differential forms and wedge products? What I have worked out so far: an $n$-form is anything that can be integrated. An example of a one form would be ...
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44 views

Confusion about orientations in Greens second identity

This question has been the source of some confusion on my part so I am hoping there is someone out there who can clear it up. Let $\Omega \in \mathbb{C}$ and $f,g\in C^{\infty}_c(\mathbb{C})$. It is ...
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30 views

Rotationally symmetric hypersurfaces with mean curvature bounded away from 0

I know that the rotationally symmetric hypersurfaces in $\mathbb{R}^n$ with constant mean curvature are the hyperplane, sphere, cylinder, catenoid, nodoid, and unduloid. Are there any significant ...
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80 views

Notation in riemannian geometry

I am reading a lecture on Riemannian geometry in which it is written that, for a differentiable manifold $M$ and a differentiable curve $v \, : \, I \, \longrightarrow \, M$ defined on an interval ...
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145 views

Use Möbius Transformation Normal Form to prove Lambda

I'm just completely lost on how to answer this question: Let $$\frac{Tz-p}{Tz-q}=\lambda \frac{z-p}{z-q}$$ be the normal form of a Möbius transformation with two fixed points. Prove that $\lambda$ = ...
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62 views

Lie Derivative Using Difference Quotient

Calculate the Lie derivative of $$G(x,y) \ = \ (x^2 \ + \ y,2y)$$ along $$F(x,y) \ = \ (2x \ + \ 3y,5x)$$ using the definition $$\mathcal{L}_F(G(\vec{x}_0)) \ = \ \ \lim_{t \rightarrow 0} ...
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198 views

About Hessian of distance function

I'm studying the comparison Hessian theorem and I not understand the following: Let $(M, \langle\ ,\ \rangle)$ be a complete Riemannian manifold. Given $o\in M$, define $r=dist(o, \cdot)$. Then, for ...
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29 views

Finite approximation of path space.

Let $M$ be a connected Riemannian manifold, $\Omega(M)=\Omega$ the path space and $E: \Omega \to \mathbb{R}$ the energy function. We can define $\Omega^c:=E^{-1}([0,c])$ and $\Omega(t_0, \dots, t_k)$ ...
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118 views

Second variation formula and Jacobi fields

Let $\bar{\alpha}:U\rightarrow \Omega$ be a two-parameter variation od a geodesic $\gamma$. For $i=1,2$ we define $$W_{i}=\frac{\partial \bar{\alpha}}{\partial u_{i}} \in T_{\gamma}\Omega$$ the ...
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24 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
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86 views

Lamé parameters and distance on a curved surface

I was wondering if it is possible to compute the distance between two points which lay on a curved analytical surface. The surface is defined with differential geometry formulae (position vector of ...
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43 views

Orthochronous Lorentz is time preserving and $\operatorname{SL}(2,R)$

Let's consider the pseudosphere in $\mathbb{R}^{1,2}$ given by $x^2+y^2-z^2=-R^2.$ We know that the group $O(1,2)=\{ A \in \operatorname{Mat}(3,\mathbb{R}): A^tGA=G \}$ where ...
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48 views

How many normal planes?

Consider a surface $S$, a point $p$ on the surface, and the unit normal vector $\vec{N}$ passing through $p$ on the surface. There are infinitely normal planes passing $p$ and its unit normal vector ...
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Asymptotic invariants of infinite groups

I am reading a Gromov's book " Metric Structures for Riemannian and Non-Riemannian Spaces ". Consider the following concept : $$ distort(X)\doteq sup \frac{length\ dist|_X}{dist|_X} $$ That is for ...
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40 views

Formalizing a proof using Germs to define a linear and injective map of the Algebraic Tangent of a manifold

I am trying to show that for X being an n-dimensional manifold and Y a k-dimensional manifold, U an open set and both $Y,U \subset X, p\in U$ there is a natural and well defined and injective map ...
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45 views

Exactness of $\Gamma^\infty$ Functor

Does anybody know a reference for fact that the Functor $\Gamma^\infty$, assigning to every smooth vector bundle $\mathcal{E}\to M$ the $C^\infty(M)$-module $\Gamma^\infty(\mathcal{E})$ of smooth ...
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98 views

What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$. The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion. ...
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70 views

Radius of geodesic disc

I'm trying to find the radius of a geodesic disc with center (0,0,1) on the paraboloid $$ r(u,v) = [\sqrt{1-u} \cos(v), \sqrt{1-u} \sin(v),u] $$ expressed by $u.$ That is, the length of the curve ...
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63 views

Operations on vector spaces

Is there a symmetrical analogue of the grassmann tensor products? Is it the polyadic tensor product? Are such symmetrical products used anywhere in differential geometry?
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74 views

Following a polyline along the surface of a polygon that is twisted

I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ...
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83 views

(Basic) question regarding Einstein-Hilbert-functional / total scalar curvature

I got a question regarding the total scalar curvature / Einstein-Hilbert-functional. I found it in Peter Topping's book "Lectures on the Ricci flow" (you can download it for free here: ...
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81 views

Proof of Lie theorem on solvable Lie algebra

I am reading a book of Helgason. As you know, solvable Lie algebra $g \subset V= {\bf C}^n$ have a nonzero $v$ such that $v$ is an eigenvector of any element of $g$. I can follow the proof in ...
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25 views

A better way to see this relation concerning Ricci tensor components

If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
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80 views

What is the needed background to study tensors?

What is the needed background to study tensors? I do not plan to study it but, I want to know the background! It's a matter of curiosity . Thanks.
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252 views

Relation between triangle and its circumscribed circle, on the surface of a sphere. Generalizations to higher dimensions.

Consider the unit sphere in $R^3$ and an equilateral triangle of side length 1, with all vertices on the surface of the sphere. Now project (from the center of the sphere outward) the triangle and its ...
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62 views

How can I align the angle between points with the magnetic heading as the points move?

I have 3 robots which must track a point. The distance between all the robots and the point is known so a triangle can be formed between any 2 robots and the point. If I find the angles in the ...
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119 views

Deformation retract

How to prove that $r_t$ is a deformation retract $M^a=\lbrace q\in M ; f(q)\leq a\rbrace$ We have the definition : $r_t$ is a difformation retract if: $r_t$ is a continius ,onto application ...
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33 views

Equivalence class involving Lie Brackets..

Can anyone help me with a proof, I spent hours on it and nothing: I want to show $$\overline{[fX, Y]}=f\overline{[X, Y]},$$ where $X$ and $Y$ are smooth vector fields on a smooth manifold $M$, $f\in ...
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70 views

Cotangent bundle of a complex projective space

How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold?