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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131 views

Local sections of Hopf fibration $(S^3,\pi,S^2)$

In the lecture we showed the local triviality of the Hopf fibration $(S^3,\pi,S^2)$ as a principal-$S^1$-bundle by constructing local sections $$s_1:S^2\setminus\{\infty\}\cong\mathbb{C}\to S^3,\qquad ...
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380 views

compute principal directions of a cylinder

I calculated the parametric equation of a cylinder, $$x(u,v)=a\cos(u)$$ $$y(u,v)=a\sin(u)$$ $$z(u,v)=v$$ I do not know how to calculate principal directions ? I am not sure what it means neither ...
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392 views

plane curves and osculating plane

Let $\alpha$ be a curve such that $|\alpha'(s)|=1$ for all $t$ and $k\neq 0$. The tangent vector $\vec T(s)$ and the normal vector $\vec N(s)$ through $\alpha(s)$ span a plane called the osculating ...
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1answer
101 views

De Rham Coohomology of Hopf Surface

How I calculate the De Rham coohomology of the Hopf surface? In particular I would like to know why the second Betti number is zero .
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1answer
127 views

Why do Zoll metrics exist only on $S^2$ and $RP^2$?

Zoll metric on a Riemannian manifold is a metric for which all geodesics are closed and have the same period. For sure, a standart metric on the sphere $S^2$ has this property: all its geodesics are ...
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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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55 views

Does $L^2$ commute with Hom?

Let $E,F \to M$ be two smooth vector bundles over a compact manifold $M$. It is well-known that the homomorphism fields $Hom(E, F) \to M$ are a smooth vector bundle, too. In fact, this bundle can be ...
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583 views

Outer Unit Normal: Cylinder

I have a cylinder occupying the region $x_{1}^{2}+x_{2}^{2} = R^2$ and $-G< x_3 < 0$ All I want to do is define the outer unit normal on the curved face. I thought about just calling it $e_1$ ...
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78 views

Finding critical points

Let $S\subset \Bbb{R^3}$ be the surface given by $x^2/4+y^2/9+z^2=1$. For $p_0=(1,0,0)$, define $f:S \to \Bbb{R} $ by $f(p)=|p-p_0|$. Then how can I find the critical points of $f$? If I use 2 ...
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27 views

Doubt on proof showing matrices of certain rank form a submanifold. [duplicate]

I have gone through two proofs to show that matrices of rank $k$, where $0 \leq k \leq \min(m,n)$ form a submanifold of the set $ M(m \times n ,\mathbb{R}) $. Both proofs involve writing down an ...
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303 views

Find isometry group of the Poincaré disc

I cannot show that any isometry of the Poincaré disc $\mathbb{H}^2$ is given by $e^{i\theta} \frac{z + c}{cz + 1}$ where $\theta$ and $c$ are constants. I know that they are indeed isometries, however ...
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65 views

Computing some differential forms using complex coordinates

I was computing some things in the Poincaré disk $\mathbb{H}^2$ in complex coordinates and then I tried to show that $\sigma_r(z) = \frac{r^2}{z}$ is an isometry. However $d\sigma_r = ...
11
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2answers
276 views

What's special about $C^\infty$ functions?

In my experience, people usually use "smooth" to mean "as smooth as I need for the upcoming proofs." Those who want to be more formal might insist on smooth meaning $C^\infty$. While the operator ...
3
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2answers
374 views

Reference request: Graduate Algebra book for self study

I have had little exposure to algebra during my undergraduate degree, covering essentially only the basics of group theory with an emphasis on the symmetries of Euclidean space, and a course on Galois ...
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1answer
178 views

derivative of a positive definite matrix

Suppose that $A$ is a positive definite symmetric matrix, specifically a Riemannian metric. Can we say anything about the sign of $tr(A^{-1}\partial_i A)$?
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48 views

Direction number of curve whose points lie on the tangents of another curve

Let $X^i = x^i +ua^i $ be parametric equations of the tangent to a curve C at $x^i$ (where $a^i$ are the direction cosines of the tangent at $x^i$ to C). Let s be the arc distance along C from a ...
3
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1answer
465 views

Dense curve on torus not an embedded submanifold

In reference to Showing a subset of the torus is dense, the responders helped show the poster that the image set $f(\mathbb{R})$ is dense in the torus. But, it's not immediately clear to me why the ...
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1answer
62 views

Prove that $g^{-1}(0)$ is a $n$-dimensional manifold.

Let $A\subset \mathbb R ^n$ be open and let $g:A\to \mathbb R ^p$ be a differentiable function such that $g'(x)$ has rank $p$ whenever $g(x)=0$. Then $g^{-1}(0)$ is an $(n-p)$-dimensional manifold. ...
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1answer
250 views

condition for curve on a sphere

Let $\alpha (t)$ be a curve such that $|\alpha'(t)|=1$ for all $t\in\mathbb R$. Assume $k(t)\neq 0$, $k'(t)\neq 0$ (whereas $k=|\alpha''(t)|$ is the curvature) and $\tau(s)\neq 0$, whereas $\tau$ is ...
6
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104 views

Derivations of the algebra of differential forms

It is well known that the interior product, the Lie derivative, and the De Rham differential are derivations of the algebra of differential forms. Does there exist other derivations of this algebra ...
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351 views

Trouble computing the shape operator.

Where have I gone wrong in the following computation of the shape operator of surface? Suppose we have a surface $M = \{(x,y,f(x,y)) \: | \: (x,y) \in \mathbb{R}^2 \}$ for some nice ...
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99 views

How to show that a given vector field is not complete in $\mathbb{R}^2$

Suppose $X=(y^2,x^2)$ is a vector field in $\mathbb{R}^2$, show that there is an integral curve starting from some point $(c,c)$ is not defined on all $\mathbb{R}$, that is, $X$ is not complete in ...
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2answers
242 views

Pullback of a Volume Form Under a Diffeomorphism.

I have an exercise here, which I have no idea how to do. Problem: Let $ U $ and $ V $ be open sets in $ \mathbb{R}^{n} $ and $ f: U \to V $ an orientation-preserving diffeomorphism. Then show that ...
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56 views

Volume forms under diffeomorphisms [duplicate]

I have an exercise and I have no idea how to do it: Let be $U$ and $V$ open sets of $\mathbb{R}^{n}$ and $f:U\rightarrow V$ an orientation-preserving diffeomorphism, then ...
3
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333 views

mean curvature is trace of second fundamental form?

My understanding was that, from the Weingarten equations, mean curvature $H$ of a surface in $\mathbb{R}^3$ satisfied $$2H = \operatorname{tr}(g^{-1} b),$$ where $g$ is the first fundamental form ...
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44 views

Completely integrable geodesic flows without any degenerate point

Are there many examples of completely integrable geodesic flows (in the sense of Liouville), with say n integrals $f_1,\cdots, f_n$ such that everywhere, the differentials $(df_1,\cdots,df_n)$ are ...
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74 views

what are the various fields in which circle is treated as infinite sided regular polygon?

What are the various fields in which circle is treated as infinite sided regular polygon? What I actually mean is , "can u suggest me some applications where circle is treated as infinite sided ...
3
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1answer
199 views

Euler Lagrange equation for harmonic maps

In the paper "The existence of minimal immersions of 2-spheres" by Sacks and Uhlenbeck the authors claim that the Euler Lagrange equation for the modified functional $E_\alpha(s) = \int_M (1 + ...
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1answer
238 views

Finding Reeb Vector Field Associated with a Contact Form

I would greatly appreciate it if you could help me with the following: I'm curious as to how to find the Reeb field $R_w$ associated to a specific contact form $w$; does one actually find $R_w$ as ...
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120 views

Computing the total curvature

Let $C$ be the curve in $\Bbb{R}^2$ given by $(t-\sin t,1-\cos t)$ for $0 \le t \le 2 \pi$. I want to find the total curvature of $C$. I found it brutally by finding the curvature $k(t)$, and then ...
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488 views

Showing how to find the vertices of the circle.

Find that the circle has four vertices. $$\gamma (t)=\langle R\cos (t/R), R \sin (t/R)\rangle$$ for $t\in [0,2\pi]$ I know the theorem: Every simple closed convex curve has atleast four ...
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64 views

Equivalent definitions of Tangent space - 2

L.S., In my book Vector Analysis by Klaus Jänich, Three different 'versions' of the Tangent space of a point $p$ at a differentiable variety are being discussed. The 'geometrical': the set of ...
3
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395 views

Showing the parametrically representation of hyperbolic paraboloid. And how to find the curves $u$ and $v$ be constant.

Show that the hyperbolic paraboloid can be represented parametrically as $$r(u,v)=\langle a(u+v), b(u-v), uv\rangle$$ Find the curves $u$ is constant and $v$ is constant. I guess I need to use the ...
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1answer
26 views

To show that $\Lambda^pL(V\rightarrow W)$ and $L(\Lambda^pV\rightarrow W)$ are not necessarily isomorphic

Let $V$ and $W$ be two vector spaces. Use $L(V\rightarrow W)$ to represent the vector space of linear map from $V$ to $W$. It is proved that $\Lambda^p(V^*)\cong (\Lambda^pV)^*$, where $\Lambda$ is ...
4
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107 views

The curve has constant torsion.

Question: Show that when the curve $c_1=c_1(t)$ has constant torsion $\tau$, the curve $$c_2=c_2(t)=-\frac{1}{\tau}N+\int_{t_0}^{t}B(u)du$$ has constant curvature $-\tau$ or $+\tau$. What I ...
3
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1answer
139 views

Covariant derivative with contravariant components derivation

I'm doing Leonard Susskind's course on General Relativity (http://deimos3.apple.com/WebObjects/Core.woa/Feed/itunes.stanford.edu-dz.19344853322.019344853324 ), and I'm stuck on a particular derivation ...
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1answer
147 views

Equivalent definitions of the tangent space

L.S., In my book Vector Analysis by Klaus Jänich, Three different 'versions' of the Tangent space of a point $p$ at a differentiable variety are being discussed. The 'geometrical': the set of ...
2
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0answers
47 views

Finding an isometry given the distance between points

If $A,B,C,D \in \mathbb{R}^n$, such that $d(A,B)=d(C,D)$, then exists an isometry $f:\mathbb{R}^n\rightarrow \mathbb{R}^n $, such that $f(A)=C, f(B)=D$. Thanks a lot for the help.
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142 views

$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$

Show that for a curve lying on a sphere of radius r with nowhere vanishing torsion, the following equation is satisfied: $$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$$ Please ...
2
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3answers
59 views

What separates rotations from other co-ordinate transformations?

I am confused about some seemingly elementary ideas. From what I have understood, a rotation is just a specific class of co-ordinate transformations. If this is true, what exactly separates a rotation ...
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1answer
202 views

Comprehensive references on partial differential equations

How do the three volumes by Taylor's "Partial differential equations" compare with the two volumes with the same title by Friedrich Sauvigny's as a reference for study? What are the good and bad ...
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455 views

The tangent space of a manifold at a point given as the kernel of the jacobian of a submersion

Let $\phi:M\to N$ is a smooth map, $q\in N$ a regular value, and $V=\phi^{-1}(q)$. I want to show that, for each $p\in V$, $T_p(V)= \mathrm{ker}(\phi_*)\subseteq T_p(M)$ (where $\phi_*$ is the ...
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61 views

Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. Show that, if $\gamma$ is a unit-speed plane curve, ...
3
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1answer
78 views

Conformal mappings of non-orientable surfaces

Is it true that for any non-orientable Riemannian surface of genus 2 there exists Conformal mapping of degree two to a projective plane? Also, is the following argument works? Given any Riemann ...
3
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1answer
244 views

For a closed plane curve, showing some inequalities.

I have a problem following : Let $\gamma:[0,T]→\mathbb{R}^2$ be a closed plane curve, i.e., a regular parametrized curve such that $ \gamma$ and all its derivatives agree at 0 and $T$. For ...
3
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1answer
138 views

If $\gamma$ is spherical, then the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Let $\gamma (t)$ be a unit-speed curve with $\kappa(t)\gt0$ and $\tau(t)\neq0$ for all $t$. Show that, if $\gamma$ is spherical, i.e., if it lies on the surface of a sphere, then ...
6
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1answer
175 views

First Pontryagin class on real Grassmannian manifold?

I wonder if real Grassmannian manifold $SO(p+q)/SO(p) \times SO(q)$ have nontrivial first Pontryagin class? I only have physics background and know really little about characteristic class theory.
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860 views

Research in differential geometry

I am an 3rd year undergrad interested in mathematics and theoretical physics. I have been reading some classical differential geometry books and I want to pursue this subject further. I have three ...
8
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2answers
131 views

How to obtain $y$

The question was written with dark-blue pen. And I tried to solve this question. I obtained $x$ as it is below. But I cannot obtain $y$ Please show me how to do this. By the way, $\gamma (t)$ ...
2
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1answer
87 views

Verify that an ellipse has four vertices.

Verify that an ellipse has four vertices. The ellipse is given by $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ And I took $$x=a\cos t$$ and $$y=b \sin t$$ for $t\in [0,2\pi]$ Please can someone help ...