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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The local chart of the embedded submanifold

Let $M$ be a $m$-dimensional smooth manifolds with boundary. $N$ is an embedded submanifold of $M$ such that $\partial N = \partial M \cap N$ and $N$ is transverse to $\partial M$, that is, for any $x ...
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55 views

Reference Request concerning Jet Bundles..

can anyone recommend me a nice reference concerning jet bundles? I've been looking for one for a long time but I couldn't find it...Thanks..
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Relative de Rahm cohomology computation for two disjoint circles embedded in R^2

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
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82 views

Geodesics on pseudo-Riemannian manifolds

Consider a Riemannian manifold $M$, with a metric $g$. We can find univocally the Levi Civita connection $\nabla$ on $M$, and so a covariant derivative $D_t$ (associated to $\nabla$) along curves. A ...
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41 views

Euler characteristic in 4 dimension

Given a 4-dimensional compact manifold (torsion free), the Euler characteristic is defined as: $E_4 = \int \epsilon_{abcd}R^{ab} \wedge R^{cd}$ with $R^{ab}$ is the curvature 2-form. Perturb the ...
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$\mathbb{RP}^3 \rightarrow \mathbb{CP}^1$ defines a principal U(1)-bundle

I have to show that the map $\pi: (x_o : x_1 : x_2 : x_3) \in \mathbb{RP}^3 \rightarrow (x_0 + i x_1) : (x_2+ ix_3) \in \mathbb{CP}^1$ defines a principal U(1)-bundle. The two standard coordinate ...
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70 views

Suggestion for a good book that explain Cartan's Moving Frame and Riemannian Geometry

I'm studying Riemannian Geometry, and I'm having a lot of trouble with the book Riemannian Geometry and Differential Dorms both from do Carmo.And I would like a book with examples, calculations, if ...
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63 views

Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
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60 views

invariance of 2-form under $SO(3)$

I'm trying to understand how to derive forms that invariant under action of some group. For example 2-form on $S^2$ and on $\mathbb{R}^3$ is very interesting for me (because I have troubles with it). ...
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66 views

What do we mean by $CP^2$ with reverse orientation?

$CP^2$ and $\bar{CP^2}$ are not diffeomorphic since they have non-isomorphic intersection forms. so why do we call the latter $CP^2$ with reverse orientation? it seems like we are not just reversing a ...
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95 views

Properties of a smooth bijection

What are the basic facts about a map $F: M \to N$ between manifolds (without boundary, we might specify) which is a smooth bijection? The map from $[0,1) \to S^1$ parameterizing the circle is a ...
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46 views

1-form on the sphere

Let $F(x,y,z)=(x,y^2,z^3)$, define a 1-form on $S^2$ by $\eta(X_p)= <F,X_p>$. Let $q=(1/2, 1/2 , 1/sqrt(2))$ and $X_q=(1,-1,0)$,$Y_q=(0,2,-sqrt(2))$. Find $d\eta(X_q,Y_q)$.Thanks.
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levels curves of polynomial equations as manifolds

Q: For which real values c is the subset $f(x) = x_{1}^{2} + x_{1}^{3} - x_{2}^{2} + x_{3}x_{4} = c$ a smooth submanifold of $\mathbb{R}^4$? Try: for it to be a smooth submanifold, $c$ has to be a ...
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79 views

finite covering space of non-orientable surfaces

Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers. A ...
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253 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha ...
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91 views

Orientation manifold, what is wrong with my argument?

As I learned, a manifold M is oriented if there exists a smooth nowhere-vanishing n-form on M. So, I am very doubting about the following construction of a n-form $\omega$ on any smooth manifold M (M ...
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50 views

$h$-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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65 views

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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28 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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89 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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201 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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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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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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127 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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115 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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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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169 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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58 views

A reference about Dolbeault cohomology

I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.
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274 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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52 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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68 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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56 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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639 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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43 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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78 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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140 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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60 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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187 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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27 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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112 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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81 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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42 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 ...