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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integral of closed differential form

This is my first course in differential forms so it might be a trivial question. If $\mu$ is a $n-1$-form on $n$-dim manifold $X$ the book uses that $$\int_X{d\mu}=0.$$ Is this expression valid for ...
3
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3answers
351 views

Transverse intersection of multiple submanifolds

Let $M$ be a smooth manifold and suppose that we have three (or more) submanifolds $N_1,N_2,N_3\subset M$. What is the right notion of "transverse intersection" of $N_1,N_2,N_3$, i.e. what is the ...
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238 views

Proving homeomorphism between surface and $\mathbb{R}^2$ minus Cantor Set

I've been working with Spivak's Differential Geometry exercises and I found myself confused with this one: "Let $C\subset \mathbb{R} \subset \mathbb{R}^2$ be the Cantor set. Show that $\mathbb{R}^2 - ...
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1answer
152 views

Diffeomorphism of open intervals in $\mathbb{R}$ with specified values

I know two open intervals on $\mathbb{R}$ are diffeomorphic to each other. My question is if I have a intervals $(a-\varepsilon,b+\varepsilon)$ and $(c-\delta,d+\delta)$, is there a diffeomorphism ...
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174 views

Exponential Map

this seems to be an easy question but I'm stuck anyway. Let $\Gamma$ be a submanifold of a Riemannian manifold $(M,g)$. Let further $U$ be a coodinate neighborhood of $\Gamma$ such that a point ...
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67 views

Why can I write the curve shortening flow system as..

Consider the family of curves $$\gamma:S^1\times [0, T)\rightarrow \mathbb R^2, \gamma(u, t)=(x(u, t), y(u, t)),$$ where $u$ parametrizes the trace of the curve and $t$ parametrizes which curve is ...
2
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3answers
203 views

Why is the diffential of a map between manifolds a map between the tangent spaces?

In the books that I have seen, given a smooth map $\phi: M \rightarrow N$ where $N$ and $M$ are manifolds, the differential at a point $x$ is defined as $d \phi_x: T_x M \rightarrow T_x N$. Why is it ...
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80 views

What is the needed background to study tensors?

What is the needed background to study tensors? I do not plan to study it but, I want to know the background! It's a matter of curiosity . Thanks.
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594 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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58 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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47 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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1answer
249 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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358 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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53 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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1answer
162 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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229 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
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40 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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1answer
58 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 ...
2
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1answer
99 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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378 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
111 views

Partitions of unity and bump function

Exercise $13.4$ $\quad$ Let $F:N\to M$ be a $C^\infty$ map of manifolds and $h:M\to\mathbb R$ a $C^\infty$ real-valued function. Prove that $\operatorname{supp}(F^*h)\subseteq ...
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79 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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92 views

A question about Möbius strip

The Möbius 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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46 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
245 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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0answers
79 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
218 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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341 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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75 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.
2
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1answer
125 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
546 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
249 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
125 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
2
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0answers
81 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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250 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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62 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
39 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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168 views

Is there a relation between Super Riemannian manifolds and Kähler 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 Kähler manifolds, or at ...
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119 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 ...
2
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2answers
73 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 ...
2
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1answer
134 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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33 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
322 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) ...
2
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1answer
72 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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275 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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0answers
113 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
73 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 ...
2
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1answer
44 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 ...
4
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1answer
158 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
84 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 ...