Tagged Questions

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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12 views

$C^{\infty}$-homotopy type of the Moebius band

The Moebius band $N$ has the same $C^{\infty}$-homotopy type of $S^1 \times \mathbb{R}$. What is the explicit expression of the $2$ $C^{\infty}$-homotopies involved ?
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0answers
11 views

Does this notion of the “directed area” of a closed curve in $\mathbb R^3$ have a standard name?

Given an oriented surface $\Omega$ in $\mathbb R^3$, consider the quantity $\mathbf A=\int_\Omega\hat n\,\mathrm dA$. We may call this the "directed area" of the surface because, when $\Omega$ is ...
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11 views

Can you give a example about of curvature tensor

Can your give a Riemann manifold $(M^n,g)$,let $R(X,Y,Z,W)=g(R(Z,W)X,Y)$,and under some coframe $w^1,...w^n$, $$R=R_{ijkl}w^i \bigotimes w^j\bigotimes w^k \bigotimes w^l$$ such that,$\forall i,j ...
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1answer
36 views

Wedge product = set intersection?

In a research article [1] I found the following formulation: The wedge product may be considered as set intersection. For example, surfaces of constant $f(x,y,z)$ and surface of constant ...
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0answers
27 views

Laplace with heaviside step function [on hold]

solve the IVP $$y''-5y'-14y=9t+u_3(t)+4(t-1)u_1(t), \quad y(0)=0, \quad y'(0)=10$$
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0answers
22 views

Relating Differential geometry with ODEs / conformal map

Let $f:\mathbb{R}_{>0} \times (0,2\pi) \rightarrow \mathbb{R}^3$ $$f(t,\phi) := (r(t) \cos( \phi) , r(t) \sin(\phi),z(t))$$ be a surface of revolution, where we assume that $r>0$ and ...
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1answer
32 views

Do closed submanifolds have nonzero curvature on a non-zero-measure set

If $M\subset \mathbb{R}^n$ is a $m$-dimensional submanifold, does the set with non-zero gaussian curvature always have $m$-dimensional Hausdorff measure greater than zero? For a manifold homeomorphic ...
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3answers
41 views

For a differentiable map $f: \mathbb{R^n}\to \mathbb{R^n}$, Show that $f^*({dy_1 \wedge\cdots \wedge dy_n})=\det(df)dx_1\wedge \cdots\wedge dx_n$

Let $f: \mathbb{R^n}\to \mathbb{R^n}$ be a differentiable map given by $f(x_1,\cdots, x_n) = (y_1,\cdots,y_n)$. Show that $f^*({dy_1 \wedge\cdots \wedge dy_n})=\det(df)dx_1\wedge \cdots\wedge dx_n$ ...
5
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1answer
92 views

What happens if you follow the sun?

Travelling around for quite a while and sometimes, well, just following the sun, today the question occurred to me: What happens if you really do this? So let's say some point is moving along the ...
3
votes
1answer
40 views

A compact connected solvable Lie group is a torus

I am looking for the proof of the following statement. A compact connected solvable Lie group of dimension $n\geq 1$ is a torus, i.e., it isomorphic to the product of $n$ copies of $S^1$. A ...
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2answers
118 views

why do you need to know topology to study differentional geometry

Why do I need to know topology to study differentional geometry? I just try to understand differentional geometry, but I am not sure why topology is needed for it. while I see that topology is an ...
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0answers
19 views

Christophel Symbols and planar

How can we get torsion from the christophel symbols? I want to show something is planar and am using christophel symbols, but how can I get torsion?
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0answers
31 views

Derivations on a Manifold

Let $M\subset \mathbb{R}^n$ be an $m$-dimensional manifold (in the ordinary Euclidean sense). Given a point $p\in M$ we define a derivation $D_p$ (at $p$) to be linear functional on the space of ...
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0answers
33 views

$\mathbb{C}P^1$ diffeomorphic to $S^2$

I am trying to show that the complex projective line is diffeomorphic to the 2-sphere. I'm using the $C^{\infty}$ structure on $\mathbb{C}P^1$ given by $U_1 = \{ [z_1 : 1], z_1 \in \mathbb{C} \}$, ...
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0answers
14 views

If a surface has no umbilic points how can it have planar points with a nonzero constant curvature k1

If we have a surface that has no umbilic points, but has a nonzero constant curvature $k_1$, how can there be any planar points? I am confused because for a umbilic point $k_1=k_2$ and a planar point ...
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1answer
17 views

Isomorphism of principal $G$-bundle

If $\pi: P\rightarrow M$ is a principal $G$-bundle we say $f:P\rightarrow P$ is automorphism of $P$ if $f$ is smooth, $f$ takes $\pi^{-1}(x)$ to $\pi^{-1}(x)$ and $f$ is compatible with $G$-action. ...
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1answer
11 views

How to find the reflection matrix

$V$ is an m dimensional subspace of $\mathbb{C}^n$ , n>m with an orthonormal basis {$q_1$,..,$q_m$}. How to find the reflector $P\in \mathbb{C}^{nxn}$ that reflects about $V$. $P$ must depend on ...
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0answers
18 views

On definition of alternating multilinear form

I am still trying to understand differential forms. I understand that locally a differential $p$-form is a alternating multilinear form $T_xM \times \dots \times T_xM \to \mathbb R$ where the domain ...
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1answer
18 views

$S = \{(x,y,z) | x^5 + y^4 + z^5 + e^z = 3\}$, why can we write $S$ as a graph $z = f(x,y)$ near (1,1,0) for smooth $f(x,y)$??

So is the whole point that at $(1,1,0)$, we have our $e^z = 1$, so then around that point we have approximately $$z = (-x^3 - y^4 +2) ^{1/5}?$$ Also we are asked to calculate $\frac{\partial ...
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0answers
11 views

Cutting a surface at critical levels produces cylinders

Let $F$ a closed surface with isolated critical points and a homeomorphism $g: F \rightarrow F$ that maps critical levels to critical levels. Let us cut the surface $F$ by the critical levels. Do we ...
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0answers
16 views

operation on non-isometric surfaces with same Gauss curvature

SameK_diffEFG Apart from above text-book example are there more $ ds^2 $ metrically paired examples? I.e., surfaces with same K(u,v) as functions of both u and v? Without agreement of first ...
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1answer
29 views

Area in the Hyperbolic Plane

Let $D_2(0) = \{(x,y) \in \mathbb{R}^2 \ | x^2+y^2 \leq 4\}$ with the Riemannian metric $$\langle u,v\rangle_{(x,y)} = \frac{u\cdot v}{g^2(x,y)} \ \ \ \ \ g(x,y) = 1 - \frac{x^2+y^2}{4}$$ I want to ...
3
votes
1answer
24 views

Boundedness Spectral Triple Axioms for de Rham Complex

In Connes' axioms for a spectral triple $(A,H,D)$, they have a representation of an algebra $A$ in bounded operators on a Hilbert space $H$, and (unbounded) operator $D$, such that $[D,a]$ is bounded. ...
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0answers
30 views

On the Chern connection

It is well kown that If $\,\bar\partial_E$ defines a holomorphic structure on the complex vector bundle $E \to X$ and $K$ is a hermitian metric on (the fibres of) $E$, there is a unique connection ...
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0answers
40 views

differential equation.

I'm stuck with this exercise. So, If someone might help me, I'll appreciate it too much. Let $U \subset \mathbb{R}^n$ be open and let $F:U\subset \mathbb{R}^n\rightarrow \mathbb{R}^n$ be a smooth ...
0
votes
1answer
14 views

Not unit-speed curve on a sphere

This question has already been asked: Curve on a sphere but is slightly different because I don´t have the hypothesis that $\alpha(t)$ is a unit-speed curve; anyway I wanted to do it by myself ...
3
votes
2answers
15 views

Index Theory: Can a closed curve around a single unstable fixed point have index $0$?

I know that a closed curve containing zero fixed points has index $0$. Is the converse also always true?
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0answers
8 views

Using the results of the local immersion/submersion theorems on manifolds

When $X,Y$ are $k$- and $l$-manifolds, we can have a function $f:X\rightarrow Y, x\in X$ such that $f$ is an immersion resp. submersion at $x$. The local immersion/submersion theorem now says: There ...
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0answers
24 views

A question about degrees of diffeomorphisms.

Let $\partial X$ be a compact boundaryless manifold and $Y$ be a connected manifold. It is known that if $f:\partial X\to Y$ is a diffeomorphism, and $f$ can be extended smoothly to $F:X\to Y$, then ...
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0answers
30 views

Space of closed parametric curve is a manifold?

I have a problem that I need help to prove, can anyone please suggest any proof? Suppose we have, closed parametric curve $f(t)=(x(t),y(t))'$ for $t\in (0,2\pi)$ (here, map is ...
2
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0answers
41 views

Proof of the existence of a partition of unity

I am trying to understand the proof that partitions of unity on a smooth manifold exist. To this end, I would like to post the proof here in my own words and kindly request that someone read it and ...
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1answer
22 views

Equivalence of definitions of torsion of a curve

The classical definition of torsion of a curve is $\tau(s)= -B´(s)\cdot N(s)$ where B is the binormal vector and N is the normal vector but I´ve seen another definition of torsion: $\tau=lim_{\Delta ...
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0answers
28 views

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces?

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces (explicitly)? A derivation is here but its conclusions seems not right? The expected ...
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0answers
9 views

About differential function between the sphere without poles and the hyperboliod of one sheet.

Let $S^2$ the unit sphere with the origin as center and $H=\{(x,y,z)\in \mathbb{R}^3:x^2+y^2-z^2=1 \}$. Denote by $N$ and $S$ the north and south pole respectively, and let ...
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0answers
18 views

Definition of the first Chern class in terms of the Ricci form

From B, B & S - String Theory and M-Theory: What does the square bracket mean? Obviously since $\mathcal{R}$ is a form and $c_1$ is a number, $[.]$ has to be an operator on forms ...
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0answers
17 views

Negative sign in the torsion of a curve

Why does the torsion of a curve has a negative sign in the formula $$\tau = -N\cdot (B)´$$ ? where N is the normal vector and B the binormal vector. My teacher didn´t explain it. I would appreciate ...
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0answers
16 views

A regular surface.

A half-line $[0,\infty)$ is perpendicular to a line $E$ and rotates about $E$ from a given initial position while its origin $0$ moves along $E$. The movement is such that when $[0,\infty)$ has ...
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1answer
21 views

How to prove that is not a regular surface.

In reference to this question: This equation define a regular surface? How can I prove that the union of this three plane is not a regular surface? Intuitively this is clear. Thanks!
2
votes
1answer
35 views

This equation define a regular surface?

Consider the function: $f(x,y,z)=xyz^2$ Its gradient is $\nabla f=(yz^2, xz^2, 2xyz)$ then the critical points are all in the sets $\{(x,y,0): x,y\in \mathbb{R}\}, \{(0,0,z): z\in \mathbb{R}\}$. My ...
1
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1answer
32 views

Cheeger constant for $S^2$

I want to calculate explicitly Cheeger constant for $S^2$, but I haven't found any sources or examples. I'm using this definition $$h(M)=\inf_A\{\frac{vol_{n-1}(\partial A)}{vol_n{(A)}}:vol_n(A)\leq ...
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0answers
23 views

Lie Bracket, Hopf Fibration, independence of choice

Let $S^3$ be the standard sphere (with the metric induced from $\mathbb{R}^4$) and $\pi\colon S^3\rightarrow \mathbb{C}P^1$ the hopf fibration. Equip $\mathbb{C}P^1$ with the Fubini-Study metric so ...
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1answer
47 views

The isomorphisms between $S^5$ and $SU(3)/SU(2)$? [on hold]

What is the precise isomorphisms between the coset $SU(3)/SU(2)$ and the five-sphere $S^5$?
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1answer
21 views

Subspace not open of a differentiable manifold

Suppose $M$ is an orientable differentiable manifold with dimension $n$. $U$ is a subspace of $M$. If $U$ is not open, is it true that $U$ also is an orientable differentiable manifold ? I need a ...
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1answer
47 views

Simple Vector Space Rotation

Given Data in the problem & notation convenstions We have 3 rotation vectors called $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$, magnitude of these vectors will give you angle of rotation We ...
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0answers
36 views

boundary of a 3-cell

Let $I^k=[0,1]^k$. I want to calculate $\partial (I^3)$ rigorously. In case of $I^2$, one can easily separate as $\partial I^2 =\partial\sigma_1+\partial\sigma_2$, where $\sigma_1=[0,e_1,e_2]$ and ...
0
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0answers
26 views

Loxodrome : found an error on wolfram MathWorld web site?

Could it be?... We find this claim on Wolfram MathWorld site http://mathworld.wolfram.com/SphericalSpiral.html The claim is that this curve (given in oblate spheroidal coordinates in the limit where ...
5
votes
0answers
50 views

What is a moduli space for a differential geometer?

A moduli space is a set that parametrizes objects with a fixed property and that is endowed with a particular structure. This should be an intuitive and general definition of what a moduli space is. ...
5
votes
2answers
57 views

About the term $-\nabla_{[u,v]}w$ in the definition of Riemann curvature tensor

As we know, in the definition of Riemann curvature tensor, we require $$ R(u,v)w=\nabla_u\nabla_v w-\nabla_v\nabla_u w-\nabla_{[u,v]}w $$ Could somebody tell me why we need $-\nabla_{[u,v]}w$ ...
2
votes
1answer
50 views

Fixed point set defined by an isometry is a geodesic

The question is asking to me prove that: Consider a fixed point set $F=\{x \in S : f(x)=x\}$ in a smooth Riemannian surface with $f:S \rightarrow S$ be an isometry. If $F$ is a smooth 1-manifold, then ...
1
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0answers
23 views

Restriction of Poisson bracket on symplectic leaf

Consier $\mathbb{R}^3$ endowed with the following Poisson bracket $$\{x_1, x_2\}=x_3, \,\,\, \{x_2,x_3\}=x_1, \,\,\, \{x_3,x_1\}=x_2.$$ Let $Z=x_1^2+x_2^2+x_3^2=r^2$ be a symplectic leaf of this ...