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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35 views
if the curvature is constant and positive, then it is on the circunference
I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the ...
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32 views
What are some geodesics of the metric $ds^2=\frac{1}{y^2}(dx^2+dy^2)$?
Ok, we have the metric $ds^2=\frac{1}{y^2}$ defined in the upper half plane $U=\{(x,y)\in\mathbb{R}^2|y>0\}$.
I know two geodesics are $x(t)=a-b\cos{t}$ and $y(t)=b\sin{t}$. What are some others? ...
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22 views
The continuity of principal coordinate system
$X$ is a $C^k$ hypersurface in $\mathbb R^{n+1}$ and $y$ is a fixed point on $X$. Can we find an orthogonal system $\{e_1(x),e_2(x),\cdots,e_{n+1}(x)\}$ on a neighborhood $U$ of $y$ such that 1. ...
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63 views
Show that the zero set of $f$ is an orientable submanifold of $\Bbb R^{n+1}$.
Suppose $f(x_1,...,x_{n+1})$ is a$ C^∞$ function on $\Bbb R^{n+1}$ with $0$ as a regular value. Show that the zero set of $f$ is an orientable submanifold of $\Bbb R_{n+1}$. In particular, the unit ...
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33 views
How to decide whether F is orientation-preserving or orientation-reversing as a diffeomorphism onto its image.
Let $U$ be the open set $(0,∞)×(0,2π)$ in the $(r,θ)$ -plane $R^2$.
We define $F : U ⊂ R^2 → R^2$ by $F (r, θ ) = (r cos θ , r sin θ )$.
How to decide whether F is orientation-preserving or ...
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40 views
Obvious Killing vectors?
What obvious Killing vectors do these metrics have?
(a) $ds^2=\frac{1}{y^2}(dx^2+dy^2)$, $-\infty<x<\infty,y>0$
(b) $ds^2=d\mathscr{X}^2+\sinh^2{\mathscr{X}}d\phi^2$, ...
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71 views
Figure $\infty$ is immersion of circle
Where can I find prove of:
Figure $\infty$ is immersion of circle.
More thanks for a prove or a function between these manifolds.
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60 views
Kähler Geodesics
Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric
$$\begin{pmatrix} ...
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1answer
32 views
The connectivity of the intersection of hypersurface and ball
$u$ is a function defined on a connected open set $\Omega$ of $\mathbb R^n$ containing $0$ such that $u \in C^2(\Omega)$ and $u(0)=0$. Consider the hypersurface $X=\{(x,u(x))~|~x\in\Omega\}$ and the ...
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48 views
Determining the embedding space:
I have seen a lot of discussion of alternate geometries for example on a sphere or hyperbolic saddle as opposed to a plane:
Has anyone consider the notion of that plane or hyperbolic saddle itself ...
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1answer
56 views
$F(x,y) = (x^2 +y^2,xy)$. compute $F^{∗}(u \, du+v \, dv)$
Let $F : \Bbb R^2 → \Bbb R^2$ be given by
If $u$,$v$ are the standard coordinates on the target $\Bbb R^2$,
compute $F^{∗}(u \, du+v \, dv)$.
$$F(x,y) = (x^2 +y^2,xy).$$
I am confused so much. I ...
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2answers
40 views
How can I show that if the second fundamental form of a surface is identically equal to zero, then the surface is a plane?
This is my question:
Let P be a plane considered as a surface in 3-space. Show that its second fundamental
form is zero. Conversely, show that if the second fundamental form of a surface is ...
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1answer
50 views
Partitions of unity and bump function
I can not image this guestion in my mind.can you give me graph and help how ı can prove this question please.
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1answer
43 views
Willmore energy of an ellipsoid
Given an ellipsoid of equation
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$
How can I calculate the Willmore energy of this surface knowing that its definition is:
...
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1answer
45 views
A question about Moebius strip
The Moebius strip (without boundary) $ S $ can be realized as a regular surface of $ R^3 $ (regular surface is meant in the sense of do Carmo's book). Using a 'manifold' language it can be proved that ...
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41 views
How do I show that $F^{∗}(dx∧dy∧dz) = ρ^{2} \sin φ dρ∧dφ∧dθ$.
I dont know how to solve. Please help me. I need to understand such types of the question for my exam studyings.
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1answer
26 views
projective hypersurface is a submanifold
I'm trying to prove that in $RP^2$, given a homogeneous polynomial $F$ of degree $k$, the hypersurface $Z(F)$ of the zeros of $F$ is smooth if $\frac{\partial F}{\partial x_0}$, $\frac{\partial ...
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57 views
Prove a torus is a 2-dimension C∞-Manifold and find its tangent space
(i) A torus is a doughnut-shaped surface in $\mathbb{R}^3$ that can be constructed as follows. Let $a > b > 0$ and consider the circle $C$ of radius $a$ in the $xy$-plane. By definition, the ...
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58 views
Pole of differential
Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$.
We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
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1answer
53 views
Finding the Riemann curvature tensor of the induced metric
The full problem is:
Let $(x,y,z)$ be Cartesian coordinates in $\mathbb{R}^3$. Let $x,y,z$ all be a parameterization of a surface $M$ in local coordinates $(u,v)$. Let local coordinates $(u,v)$ be ...
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122 views
Does Differential Topology or Differential Geometry play a larger role in Chaos Theory?
I'm an undergraduate on somewhat of a time constraint in school. I have room in my remaining schedule for a semester of either Differential Geometry or Differential Topology.
I understand the ...
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1answer
60 views
Find a $1$-form $ω$ on $\mathbb R^2 −\{(0,0)\}$ such that $ω(X) = 1$ and $ω(Y) = 0$.
Please ı dont know what I need to do. thus, help me to solve.
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1answer
79 views
When is a topological space a manifold?
I'm looking for someone to point me in the direction of papers or books which discuss when a topological space (perhaps with the conditions locally compact, Hausdorff) is a topological manifold, ...
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1answer
50 views
Show that this is a diffeomorphism
I have a function $F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2$
with $(\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi)))$ and want to show that this is a smooth(meaning ...
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2answers
125 views
Relationship between trace of a linear map and the number of points it fixes.
Problem Statement: Let $\Phi_A:T^2\rightarrow T^2$ be a smooth mapping into the torus induced by a linear map $A\in SL_2(\mathbb{Z})$ under the quotient relation that identifies 0 and 1. Assume that A ...
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1answer
33 views
Nondegenerate critical point
I don't understand this part from the book of Zeidler , can someone help me to understand it ?
Please
Thank you
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0answers
28 views
Curvature form projective spaces
Let $T\mathbb{C}P^n$ tangent bundle over $\mathbb{C}P^n$. We have an hermitian metric on $T
\mathbb{C}P^n$ defined as $h=\frac{dzd\bar{z}}{(1+|z|^2)^2}$. If we consider Levi-Civita connection we can ...
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24 views
Relation between triangle and its circumscribed circle, on the surface of a sphere. Generalizations to higher dimensions.
Consider the unit sphere in $R^3$ and an equilateral triangle of side length 1, with all vertices on the surface of the sphere. Now project (from the center of the sphere outward) the triangle and its ...
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21 views
How can I align the angle between points with the magnetic heading as the points move?
I have 3 robots which must track a point. The distance between all the robots and the point is known so a triangle can be formed between any 2 robots and the point.
If I find the angles in the ...
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1answer
33 views
prove that $supp(π^{∗} f) = (supp f)×N.$ Please can you check my answer? Also more explanation please.
My question is that
Let $f \colon M \to R$ be a $C^{\infty}$ function on a manifold $M$.
If $N$ is another manifold and $π \colon M \times N \to M$ is the projection onto the first factor, prove ...
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14 views
The most general form of the metric for a homogeneous, isotropic and static space-time
What is the most general form of the metric for a homogeneous, isotropic and static space-time?
For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
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39 views
Is there a relation between Super Riemannian manifolds and Kahler manifolds?
(This question has a physics motivation).
Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kahler manifolds, or at ...
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0answers
43 views
Deformation retract
How to prove that $r_t$ is a deformation retract
$M^a=\lbrace q\in M ; f(q)\leq a\rbrace$
We have the definition :
$r_t$ is a difformation retract if:
$r_t$ is a continius ,onto application ...
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1answer
36 views
Why is $\theta \not \in C^{\infty}(S^1)$?
Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
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30 views
Mymultiple image geometry
I have to work with multiple aerial images. the objective is to reconstruct 3d features.
For a particular object, i want to find the images which are giving good viewing geometry than others. so ...
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1answer
63 views
Visualize soliton solutions of a PDE
In trying to visualize soliton solutions of a PDE I faced this sentence:
We now think of solitons as self-similar solutions, i.e., solutions which evolve along
symmetries of the flow.
Question 1: ...
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0answers
23 views
Equivalence class involving Lie Brackets..
Can anyone help me with a proof, I spent hours on it and nothing: I want to show $$\overline{[fX, Y]}=f\overline{[X, Y]},$$ where $X$ and $Y$ are smooth vector fields on a smooth manifold $M$, $f\in ...
2
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1answer
56 views
Incomplete vector field
Is there a way I can tell if a vector field on a manifold or just $\mathbb{R}^n$ is incomplete simply by just looking at its formula. For instance on $\mathbb{R}$, $\displaystyle X= (x^2+1) ...
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0answers
22 views
Laplacian on Reductive coset spaces
Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$) and let $X_i$ denote the vector fields which generate rotations about the $x_i$-axis. My questions are:
(a) Is it true that ...
2
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1answer
42 views
Many partitions of unity on sufficiently “nice”; what does this mean?
In a class I am taking, we are told that are manifolds have "many" partitions of unity (we assume paracompact, Hausdorff, second-countable). However, the course content is not related to this subject. ...
2
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0answers
46 views
Chern classes tangent bundle.
I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this ...
2
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50 views
Geodesics and Christoffel symbols
If these are satisfied then we are on a geodesic. Do I just need to plug in the condition given about the christoffel symbols and then see that the equations are allways fulfiled as long as $v=at+b$ ...
4
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1answer
47 views
How to make a $C^1$ knot into a $C^\infty$ knot
Suppose I have a $C^1$ imbedding $f: S^1 \rightarrow S^3$. From the point of view of knot theory, what's the "best" way to get a $C^\infty$ curve that "looks like" or is "equivalent to" $f$? For ...
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1answer
25 views
“WLOG” when studying Schwarzschild geodesics
Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$? I assume it is so because when digging around the internet, most references seem to consider this ...
2
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1answer
39 views
Is Whitney sum of vector bundle a categorical colimit?
We known that the direct sum of two vector spaces is the categorical colimit of vector spaces. My question is whether Whitney sum of vector bundle is a categorical colimit (in the category of vector ...
2
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1answer
43 views
Help with algebraic manipulation to prove that $\Omega (M)$ constructed from $\Omega^1 (M)$ forms an algebra over $C^\infty (M)$
In Baez´s Gauge Theories, Knots and Gravity he states that the differential forms on a n dimensional manifold M, $\Omega (M) = {\bigoplus}_p \Omega^p (M)$, constructed from $\Omega^1(M)$ and the ...
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1answer
67 views
“Completing” a vector field on a non-compact manifold $M$
Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete.
Is there a way to create a smooth vector field $V$ that is ...
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1answer
25 views
Curvature (Gaussian) of a hypersphere
I am looking for a general formula for the Gaussian curvature of an $n$-sphere (the set of points in $R^{n+1}$ equidistant from the origin) of radius $r$.
From what I have read, there would be $n$ ...
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2answers
38 views
What is the initial reason to define the evolute of a curve?
The evolute of a curve is defined as the envelope of the normals or as the locus of the center of the osculating circle.
What is exactly "the envelope of the normals" ?
What is the reason to ...
4
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1answer
58 views
Differential of smooth function on manifold
In the book I am using, the author defines differentials in the following way.
Given smooth manifolds $M,N$ and a smooth mapping $\psi:M\to N$ define the differential $d\psi_m$ at a point $m\in M$ as ...
