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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129 views

immersion of punctured torus in plane

Let $S^1 \times S^1$ be the $2$-torus. If a point $a=(p,q)$ of the torus is removed, i.e., it is punctured at one point then how can I show that it can be immersed in the plane, i.e., in $\Bbb{R}^2$? ...
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57 views

Green's theorem via Stokes's theorem

I am considering the following form of Stokes's theorem: Let $\omega$ be an $n-1$ differential form with compact support on an oriented manifold of dimension $n$. Let us consider the boundary $\...
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1answer
43 views

How to proof that bracket of two vector field can be computed by second derivation

Can some one give a hint how can I proof that where $\phi$ indicated the flow of vector fields.
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1answer
55 views

Why not every homogeneous manifold is parallelizable?

It is obvious that not every homogeneous manifold is parallelizable (take for example the two-sphere $S^{2}$). In contrast, every Lie group $G$ is parallelizable, as you can construct a pointwise ...
2
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1answer
77 views

Compute Euler characteristic of a compact manifold

We have the manifold embedded in $\mathbb{R}^4$ given by $$M=\{(x,y,z,w)|2x^2+2=2z^2+w^2,3x^2+y^2=z^2+w^2\}$$ How could I compute the Euler characteristic? I've no idea computing the homology group of ...
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2answers
194 views

Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can take $...
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0answers
61 views

Pullback 1-differential form

Let $(x_0,x'_0) \in \mathbb{R}^2$ be initial data for the Euler Lagrange equation with some given Lagrangian $L: \mathbb{R}^2 \rightarrow \mathbb{R}.$ Then $F^t$ is the flow that maps the initial ...
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1answer
148 views

Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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1answer
48 views

Proving that these curves intersect

Let $\Gamma$, $\Sigma$ be two curves with ranges in $(\{0\}\cup\mathbb{R}_{+})^2$. $\Gamma$ starts on the $y$ and ends on the $x$ axis: $\Gamma(0)=(0,\gamma_2),\Gamma(1)=(\gamma_1,0)$. $\Sigma$ is a "...
2
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1answer
54 views

Question about parallel displacement on a surface

This is Problem 9.6(1) from the book The Geometry of Physics: What's wrong with the following argument? A vector $\mathbf v$ is parallel displaced around a small closed curve $C = \partial{U^...
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3answers
376 views

Möbius strip as a non-trivial principal bundle

There is a well-known theorem that a principal bundle is trivial if and only if it admits a global section. I'm trying to get a good picture of what this theorem means. The Möbius Strip can be ...
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35 views

On critical values of a linear function $g:SO_2\times \mathbb R^2\to \mathbb R^2$

Consider map $g:SO_2\times \mathbb R^2\to \mathbb R^2$, $g(A,v)=Av$, where $A\in SO_2$ is an orthogonal $2\times 2$ matrix and $v\in \mathbb R^2$ is a $2$-vector. Show that $0$ is a critical value. ...
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35 views

Self-adjointness of $X^2$

Let $X$ be a vector field on a compact (may be even complete) Riemannian manifold without boundary. I am wondering if $X^2$ will be a self-adjoint operator on $L^2(M)$. Any hints would be appreciated. ...
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51 views

Unbounded Geodesics and Nonpositive Curvature

I have the following interesting(?) question: Let $M$ be a complete Riemann manifold. Suppose that it is of non-negative curvature. I want to know if $M$ has unbounded geodesics. As the question is ...
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19 views

Fiber summation of symplectic Lefschetz fibrations.

Recall the following standard result. Theorem: For any genus $2$ symplectic Lefschetz fibration $f:X\to S^2$, there exists an integer $n_0$ such that, for all $n\ge n_0$, $f\# nf_0$ is isomorphic ...
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47 views

Total curvature of an ovaloid.

I have the following exercise that I have to solve without using Gauss-Bonnet theorem. We say that a compact surface $\Sigma \subset \mathbb{R}^3$ is an ovaloid if the Gaussian curvature $K(p)>0$ ...
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62 views

Write down the explicit form of the $15$ Killing vectors of the 5-sphere

I am looking for a way to write down explicitly the $15$ vectors which are generators of $SO(6)$ in polar coordinates on the $5$-sphere. In particular I have the round metric $$g_{\mu\nu} = \left( \...
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1answer
48 views

Jets and vertical differential

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: \begin{equation}...
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25 views

Non-affinely parametrized geodesics

Consider a non-affinely parameterised geodesic, i.e., a geodesic whose tangent vector field obeys $\nabla_X X = fX$ for some function $f$. Prove that one may reparameterise the geodesic so the tangent ...
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89 views

How to prove that the tangent map $T\phi$ into the pullback bundle is smooth?

Assume $\phi: M\rightarrow N$ is smooth. Let $\phi^*(TN)$ be the pullback bundle of $TN$ by $\phi$. Define $T\phi:TM\rightarrow \phi^*(TN)$ as follows: $T\phi(m,v)=(m,d\phi_m(v)) $. We also have the ...
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47 views

What does a skew second fundamental form geometrically mean?

Could there be a realizable 2-surface in some higher dimensional non-Riemannian embedding space whose second fundamental form is skew? If yes, then what would its skew part mean?
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63 views

Asymptotic geodesic on hyperboloid.

Consider a geodesic which starts at a point $p$ in the upper part $(z>0)$ of a hyperboloid of revolution $x^2+y^2−z^2=1$ and makes an angle $\theta$ with the parallel passing through $p$ in such a ...
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1answer
246 views

Different definitions of differential forms?

I am a physicist and was reading about differential forms in Classical Mechanics. Now, I thought that a two-form is a smooth map $\omega : M \rightarrow \Lambda(T^*M)$ so that a point $p$ on the ...
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52 views

Bianchi Identity - Gauge Theory

I am reading through some lecture notes (found here) and following a proof of the Bianchi identity in the context of principal bundles. That is, $h^*\Omega = 0$, where $\Omega$ is the curvature 2-...
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1answer
70 views

Curve Orientation on a Surface

Let $S\subset\mathbb{R}^3$ be an orientable surface, but not oriented (yet). Let $X:(u_1,u_2)\in U\subset \mathbb{R}^2\longrightarrow X(U)\subset S$ be a local parametrization of the surface $S$. We ...
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117 views

product metric on Riemannian manifolds

Let $M_1$ and $M_2$ be Riemannian manifolds and consider the cartesian product $M_1 \times M_2$ with the profuct structure .Let $\pi_1: M_1 \times M_2 \to M_1$ and $\pi_2: M_1 \times M_2 \to M_2$be ...
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1answer
155 views

Differential forms - looking for 3 definitions!

I am sorry for this type of question, but I currently have to deal with differential forms although I have not heard so far what they actually are, so I have just a few very particular questions about ...
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1answer
42 views

Parabolic points and curvature.

I have problems to solve this exercise: Let $p$ be a point of an oriented surface $S$ and assume that there is a neighborhood $U$ of $p$ in $S$ all points of which are parabolic. Prove that the (...
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33 views

Frenet frame and tangent space.

Let $\gamma(s) \subset \Sigma \subset \mathbb{R}^3$ a parametrized curve by arc lenght. Let suppose that $\gamma$ is on an oriented surface $\Sigma \subset \mathbb{R}^3$. We can consider the Frenet ...
2
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1answer
326 views

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$.

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$. I've been looking for an diffeomorphism between a sphere in $\mathbb R^3$ and an ellipsoid of the form $$\{ (x,y,z) \in \mathbb R^3 \...
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49 views

Deformation theory in a paper of Bogomolov and Tschinkel

I am trying to read this paper http://www.math.nyu.edu/~tschinke/papers/yuri/00ajm/ajm.pdf, by Bogomolov and Tschinkel. I had 0 preparation in the theory of deformation of complex structures, but in ...
3
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1answer
112 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ submanifold....
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1answer
118 views

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface?

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface ? Consider a regular surface $S \subset \mathbb R^3$ and a diffeomorphism $F: ...
4
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0answers
105 views

Metric tensors and linear (differential) operators defined on a manifold and its osculating sphere

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
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1answer
54 views

Gauge transformation on a principal bundle

I am reading through lecture notes found here and on pg 11 they define a map $\overline{\phi}_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow G$ by $\overline{\phi}_{\alpha}(p)=g_{\alpha}(\Phi(p))g_{\alpha}(...
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41 views

When the associated bundle to a $U(1)$-bundle is an holomorphic line bundle?

Let $P$ be a principal $U(1)$ bundle over a complex manifold $M$, and let $\rho\colon U(1)\to Aut(\mathbb{C})$ be the representation of $U(1)$ on $\mathbb{C}$ given by the standard multiplication. My ...
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1answer
49 views

Chronology condition and metric perturbations

Let $(M,g)$ be the quotient of the 2-dimensional Minkowski space-time by the discrete group of isometries generated by the map $f(t, x) = (t + 1, x + 1)$. Show that $(M, g)$ satisfies the chronology ...
2
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1answer
70 views

How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
4
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1answer
122 views

(co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
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3answers
28 views

How do I prove that there are only two possible orthonormal basis?

Let $x,y\in\mathbb{R}^3$ be unit vectors such that $\{x,y,(0,0,1)\}$ is an orthonormal subset of $\mathbb{R}^3$. Let $z\in \mathbb{R}^3$ such that $\{x,y,z\}$ is an orthonormal subset of $\mathbb{R}^...
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1answer
29 views

Partion the boundary of a $n$-dimensional ball and write each partition as the graph of a $C^1$-function on a open subset of $\mathbb{R}^{n-1}$

Let $$S:=\left\{x\in\mathbb{R}^n:\left\|x\right\|_2\le r\right\}$$ How can we partition the boundary $\partial S$ of $S$ and write each partition as the graph of a continuously differentiable function ...
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98 views

Correspondence between line bundles and $U(1)$-bundles: a mistake from the physicists?

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If $e^{...
3
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1answer
93 views

The Poincaré dual of a space-time curve

We have a smooth space-time curve defined by $f:C{\mapsto}M$, where $M$ is a typical curved space-time manifold. ${\eta}^{(4)}$ is the volume 4-form defined on $M$ and ${\varepsilon}^{(1)}$ is the (...
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1answer
25 views

Is $|K/\tau|\leq 1$?

Let $\alpha:[a,b]\rightarrow \mathbb{R}^3$ be a regular curve such that $\alpha'$ and $\alpha''$ are linearly independent over $[a,b]$. Let $K$ and $\tau$ be the curvature and torsion of $\alpha$ ...
3
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1answer
184 views

Classification of line bundles by Griffiths and Harris

I am reading pages $132$ and $133$ of Principles of Algebraic Geometry by Griffiths and Harris. They consider a holomorphic line bundle $L \to M$ over a manifold $M$ and an open cover $\left\{ U_{a}\...
3
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1answer
163 views

Ricci curvature of the Grassmannian?

Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space $(\mathbb{P}^N,\...
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1answer
78 views

Constant torsion-expression of unit speed curves

I am currently studying for an exam in differential geometry. There's a problem which I am not able to solve and do not even know where to start (although I think it has to do with the Frenet ...
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1answer
80 views

Bound on the surface integral of the absolute Gaussian curvature given by surface of unit sphere?

im currently looking at an application in which I have to calculate or at least approximate the surface integral of the absolute Gaussian curvature over a patch of a regular surface $S$ given by a ...
4
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1answer
58 views

Can some one help me parametrize $\frac{x^4}{a^4}+\frac{y^4}{b^4}+\frac{z^4}{c^4}=1$

Given a surface $$\frac{x^4}{a^4}+\frac{y^4}{b^4}+\frac{z^4}{c^4}=1$$how can I parametrize the surface using $X(u,v).$ I tried to use $$x=a\sqrt{\cos(\theta)\sin(\phi)}$$ $$y=b\sqrt{\cos(\theta)\sin(\...
2
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1answer
147 views

How can I prove $dz=dx+idy$?

Let's see $\Bbb C$ as an $\Bbb R$-vector space. Hence it is isomorphic to $\Bbb R^2$ and it has dimension $2$. If $v_1,v_2$ is a basis for $\Bbb R^2$, every its element can be written as $xv_1+yv_2$; ...