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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homeomorphism still isotopic to the identity after the deletion of points.

I have a surface $M$ (without boundary) and a homeomorphism $f:M \rightarrow M$, which is isotopic to the identity on $M$. If I delete two points $x$ and $f(x)$ from the surface, I get a ...
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30 views

Obstruction of such gauge choice

Suppose we consider $\operatorname{ad}P_G \to T^k$ as the associated adjoint bundle (maybe this is not the correct name, but I just mean with the associated vector bundle ${\rm Lie}G$ as standard ...
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38 views

Orientability of smooth structures on the real line

How I can prove that every every smooth structure on the real line (with the standard topology) is orientable?
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58 views

prove that the shapes are isometric

I want to prove that the shapes are isometric. How to prove? There is no info except for the picture. First of all I need to write surface patches. Please can someone help me? The definition of ...
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31 views

Existence of slices for the action of a subgroup

Assume that a group $G$ acts on a space $M$ in such a way that there exists a slice at a point $m \in M$. Let $H \subseteq G$ be a subgroup. Under which additional assumptions (if there are any) can ...
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128 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
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54 views

Differentiating the exponential map of a unitary group

I consider a unitary group $U \in U(N)$ with an exponential map: $ U = \exp (iH)$, with $H$ hermitian. I am not sure if I can do the following calculation: $U^\dagger d U = \exp (-iH) i dH \exp (iH) ...
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41 views

Holder estimate for eigenfunctions of Laplace operator on sphere?

I would like to ask if there is a holder esitimate for eigenfunctions of Laplace operator on sphere? I mean the esimate for \begin{equation} \|\partial^{\alpha}h_n\|_{L^{\infty}(S^{d-1})} ...
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show that $g\circ f :S_1 \to S_3$ be also smooth and $d_p(g\circ f)=(d_{f(p)}g)\circ (d_pf)$

proposition: Let $S_1$, $S_2$, $S_3$ be 3-regular surfaces and $f:S_1 \to S_2$ and $g: S_2 \to S_3$ be smooth maps. Then show that $g\circ f :S_1 \to S_3$ be also smooth and $$d_p(g\circ ...
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110 views

$k$-jets of sections of a vector bundle..

I need some help for establishing a connection between two definitions of $k$-jets: Algebraic Definition: Let $E\rightarrow M$ be a smooth vector bundle and define the ideal of $\Gamma(E)$: ...
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53 views

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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57 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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50 views

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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91 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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36 views

$\mathbb{RP}^3 \rightarrow \mathbb{CP}^1$ 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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72 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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64 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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99 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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47 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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80 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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257 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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53 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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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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$S^1$-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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93 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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218 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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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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132 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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180 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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61 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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280 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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69 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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661 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 ...