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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Show that a k-form can be expressed as wedge product

I'm trying to show that given a $1$-form $\omega$ and a $k$-form $\alpha$ such that $\alpha \wedge \omega = 0$ then there exists a $(k-1)$-form $\beta$ such that $\alpha = \omega \wedge \beta$. I'm ...
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117 views

Two nonevident implications in a proof

I am reading part of Lee's introduction to mainfolds. I got to the following proposition. $\textbf{Proposition 14.7.}$ Suppose $H\subset N$ is an integral manifold of an involutive distribution ...
6
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1answer
77 views

Why can't that be an uncountable union?

I'm reading part of Lee's Introduction to manifolds. I have come to the following proposition. $\textbf{Proposition 14.6 (Local Structure of Integral Manifolds).}$ Let $D$ be an involutive ...
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246 views

The quotient of a manifold by a submanifold is never a manifold?

Let $M$ be a connected smooth manifold. Let $S$ be a connected embedded submanifold of positive dimension and co-dimension, which is also a closed subset of $M$. Is it true that the quotient space ...
2
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1answer
52 views

Proving this result on tangent spaces to foliations

Reading through Lee's introduction to smooth manifold, I bumped into this result: $\textbf{Lemma 14.12.}$ Let $\cal F$ be a foliation of a smooth manifold $M$. The collection of tangent spaces to ...
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68 views

Problems understanding this proof

$\textbf{Theorem 3.5.10}$ (Arnol'd $[1]$). Suppose $(M,\sigma)$ is a symplectic manifold of dimension $2n$, let $f_1,...,f_n$ be an involution on $M$, and finally assume that the Hamilton fields ...
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2answers
112 views

Show $f$ is an isometry from $s$ to $s'$

Let $s$ denote the surface of revolution $$(x,y,z)=(\cos \theta \cosh v, \sin \theta \cosh v,v)$$ where $0 < \theta < 2 \pi$ and $-\infty < v < \infty$ Let $s'$ denote the surface ...
2
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2answers
79 views

de Rham cohomology on finitely smooth manifolds

In all of the places I've looked, de Rham cohomology is defined on $\mathcal{C}^\infty$ manifolds with $\mathcal{C}^\infty$ differential forms. What about de Rham cohomology on $\mathcal{C}^r$ ...
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1answer
38 views

Is a translation in a compact Lie group homotopic to the identity?

The following exercise is from Guillemin and Pollack, Differential Topology. Show that the Euler characteristic of the orthogonal group (or any compact Lie group for that matter) is zero. Hint: ...
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48 views

Proving those are manifolds

Reading through McDuff, Salamon, I came across the following extract: ing geometrically. Because the functions $F_i$ are constants of the motion, the level sets $T_c=\{z\in \Omega|F_j(z)=c_j\}$ ...
3
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240 views

Intuition behind variational principle

Hofer-Zehnder, in section 1.5, proves that every Hamiltonian field on a strictly convex compact regular energy surface carries a periodic orbit. I have understood the proof. What I am wondering about ...
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93 views

What's this “Euler's formula”?

convex. Indeed, differentiating Euler's foruma $\langle F_x,x\rangle=F$ gives $F_{xx}x=0$. By the convexity assumption $F_{xx}\,\Big|\,TS>0$. Therefore, we define the function From ...
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Metric with scalar curvature-1

i have this simple question : can someone give me the definition or an indication to know what is the "Metric with scalar curvature-1 " thanks .....
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23 views

Easier way to induce an orientation to the border of a manifold

I'm working in the following exercise: Let $M=\{(x, y, z): x²+y²=1 \,\text{and}\, 0\leq z \leq 1\}$. Let $\alpha:(0,1)²\rightarrow \mathbb R³$ be given by $\alpha(u, v)=(\cos u, \sin u, v)$. ...
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1answer
18 views

Showing that, at an elliptic point, a surface lies on one side of the tangent plane.

Let $p\in S$ be an elliptic point of a surface $S$. I want to show that there exists a neighbourhhod $V$ of $p$ in $S$ such that all points in $V$ belong to the same side of the tangent plane ...
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23 views

Raising index on covariant derivative

So suppose $X$ is some vector field and $t$ is a tangent vector to some curve on some smooth manifold. Then $t^a\nabla_a X$ gives the directional derivative of the vector field in the direction of ...
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43 views

basic question of differential geometry.

I'm taking a course on general relativity and I did a problem that I have now found out I have no idea how to interpret. The surface of a paraboloid has the metric $$ds^2=(1+r^2)dr^2+r^2d\theta^2$$ ...
4
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479 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
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27 views

On Automorphism on Space of Smooth Functions

Given a Operator $T$ (an Automorphism) on the subspace $X$ of Smooth functions on $\mathbb R ^n$, $\mathcal C^\infty(\mathbb R^n)$ $$ X=\{u \in \mathcal C^\infty (\mathbb R^n) \,|\, supp ...
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17 views

Symmetric Matrix for Shape Operator?

Let $R$ be a smooth surface (smoothly embedded) in $\mathbb{R}^3$. Let $M$ be the matrix for the Shape operator of $R$ with respect to the basis $\{\partial _x F, \partial_yF\}$ for the tangent space ...
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9 views

Metric of the doubling i.e circular billard table

Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming ...
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467 views

Non-vanishing differential form: what does it mean?

A $1$-form $\alpha$ over a smooth manifold is non vanishing if for every $p\in M$, $\alpha_p\neq 0$. But $\alpha_p$ is a linear map $T_p M\to \mathbb R$ hence $\alpha_p(0)=0$. So confusion arises ...
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12 views

parameterization of surface of revolution?

parameterization of surface of revolution formed by revolving the $x=\cosh z$ around z axis , i thought the it as $$x=\cosh z \cos \theta ,y=\cosh z \sin \theta ,z=z$$ Hence the surface can be ...
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17 views

Showing that a mapping is an isometry

Can anyone please help with this question? I have tried substituting but I'm not sure if that's correct so think I am missing something. Let S denote the surface of revolution $(x, y, z) = ...
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33 views

Surface area of Convex bodies contained in one another

Suppose we have two compact convex bodies one contained in the other in $\mathbb{R}^n$, $C\subset D\subset \mathbb{R}^n$. Does it follow that the ($n-1$ dimensional) surface area of $C$ is less than ...
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22 views

How do I show that the reparametrization of a pre-geodesic is pre-geodesic?

So this is probably a very silly question but I need to prove the reparametrization of a pre-geodesic is a pre-geodesic. The hint in my book says it's obvious, so I am clearly missing something.
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1answer
37 views

Dimension of topology manifold

In the 3 page of Jurgen Jost's Riemannian Geometry and Geometric Analysis .Why it is harder in topology manifold than differentiable manifold ? I think it is easy in differentiable manifold because ...
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1answer
45 views

Curvature of the Involute

I'm trying to calculate the curvature of the involute of an arbitrary, not necessarily unit speed curve, and show that it can be written in terms of the curvature, torsion, and arclength function of ...
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28 views

Proper patch in the differential geometry

I have a question that coincides with this question. Proving that every patch in a surface $M$ in $R^3$ is proper. Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries ...
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1answer
30 views

Every compact hypersurface in $\mathbb{R}^n$ is orientable

Show that every compact hypersurface in $\mathbb{R}^n$ is orientable. HINT: Jordan-Brouwer Separation Theorem. This is an exercise from Guillemin and Pollack. So hypersurface means smooth ...
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1answer
28 views

Definition of Levi-Civita connection map?

Does anyone know definition of Levi-Civita connection map that defined as $TTM\to TM$. I would appreciate if you could give a good reference. Thanks.
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22 views

Example of a Symbol: Connection.

I'm trying to get more intuition for the symbol of a differential operator. In particular, I've tried the example of a connection. What is the most efficient way to compute the symbol for, say, a ...
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15 views

Finding a zero of homogeneous parts of polynomials using zero of those polynomials

Let $f, g \in \mathbb{Q}[x_1, ..., x_n]$ be polynomials of degree $d$. Let $F$ and $G$ denote the degree $d$ portions of $f$ and $g$ respectively. Suppose there exists a non-singular point ...
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0answers
22 views

Proving the invariance of the mixed product under direct congruent transformations

Text: Differential Geometry, by Erwin Kreyszig Given a set of vectors: $$ \overrightarrow{a}=\langle a_1, a_2, a_3 \rangle $$ $$ \overrightarrow{b}=\langle b_1, b_2, b_3 \rangle $$ $$ ...
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1answer
100 views

Show that the convex neighborhood in a Riemannian Manifold are subset contractibles

Show that the convex neighborhood in a Riemannian Manifold are subset contractibles (to any of their points). A subset $A$ of the differential manifold $M$ is contractible to the point $a\in A$ ...
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1answer
36 views

Lie Groups map question

This is a question about Exercise 5.59 in Jeff Lee's Manifolds and Differential Geometry. He writes For a fixed $A\in GL(V)$, the map $L_A:GL(V)\rightarrow GL(V)$ given by $A\mapsto A\circ B$ has ...
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1answer
44 views

Differential forms on a point

For the proof of Poincaré lemma, it's essential to evaluate $\Omega^p(*)$ where $*$ is zero dimensional manifold and $\Omega^p$ is a collection of all $p$-forms on given manifold. Clearly, $\Omega^0 ...
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2answers
58 views

Smooth Manifolds and Lie Group Action

I have started to read about Lie group action on smooth manifolds. A question popped up in my mind and I am not sure it's silly or not. Diff (M) = space of all smooth diffeomorphisms is a large ...
4
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1answer
107 views

Group Action and Smooth Manifolds

I was wondering if it is sufficient for a compact (i.e. Hausdorff) smooth manifold $M$ to have a free group action of a finite group $G$ in order to conclude that $M/G$ is a compact smooth manifold? ...
3
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1answer
479 views

Smooth Manifold with Trivial Tangent Bundle

So, I'm a little confused about one statement made in class today : If M is a smooth manifold without boundary such that the tangent bundle of M is trivial, then M is orientable. Is this ...
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2answers
77 views

Algebraic Varieties vs Smooth Manifolds

There are many posts I have read on that subject which seem unclear for me. My main question (it might be silly) is: "Every non-singular algebraic variety over $\mathbb{C}$ is a smooth ...
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2answers
42 views

Exact vs. conservative

I'm having trouble understanding definitions. What's the difference between something being exact and being conservative? I understand both involve proving that a potential function $f$ exists such ...
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1answer
262 views

How to Prove this Map is an Embedding of a Smooth Manifold?

I have run into a problem in my differential geometry book. Let $M$ be a smooth manifold and $F={C^\infty }(M,\mathbb R)$. Define a mapping $i:M \to {\mathbb R^F}$ by ${i_f}(x) = f(x)$ for ...
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2answers
543 views

Why is the $n$-sphere a Smooth Manifold?

I want to know how the $n$-sphere is a smooth manifold? I'm unable to understand the theory of differentiable structure on the $n$-sphere. Please tell me any suggested reading for a good start on ...
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1answer
88 views

Is the Intersection of these Two Sets a Smooth Manifold?

$A=M\cap N$, $$M=\{(x,y,z)\in\Bbb R^3| x^2+y^2=1\},$$ $$N=\{(x,y,z)\in \Bbb R^3|x^2-xy+y^2-z=1\}.$$ 1. Is $A$ is smooth manifold? 2. Find the points of $A$ that are farthest from the ...
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Proof that the change of parameters between two regular surfaces is a diffeomorphism

I'm stuck on the proof that the change of parameters between two regular surfaces is a diffeomorphism. I'm using Do Carmo's book Differential Geometry of Curves and Surfaces, which can be found online ...
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1answer
29 views

Find the curvature tensor and sectional curvature associated with the first fundamental form$I=du^2+f^2(u)dv^2$

Consider the surface of revolution $\sigma$ in he Euclidean space $\mathbb{R^3}$ given by $$\sigma(u,v)=(f(u)cosv,f(u)sinv, g(u))$$ with $f>0$ where the profile curve has unit speed. The first ...
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1answer
16 views

The bundle of spin frames as an associated bundle

$\DeclareMathOperator{\Spin}{Spin}$Let $X$ be an oriented smooth $n$-manifold with the frame bundle $\pi_{SO} \colon F_{SO} \to X$. Then the bundle of spin frames is a $\Spin(n)$-bundle $\pi_{\Spin} ...
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1answer
28 views

Level set of the hopf map

The Hopf map is given by the projection $\pi: \mathbb{S}^3 \to \mathbb{S}^2$, and: $\pi: z \mapsto zi\bar{z}$, where $z \in \mathbb{S}^3$ and $i \in \mathbb{H}$ is a unit quaternion. Show that ...
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calculation of laplacian- beltrami operator on sphere in terms of (x,y,z)?

As the answer given by @mark fischler in my previous question How to change metric variable to x,y,z The metric on sphere could be written in terms of x,y,z could be written as $(ds)^2$: $$ (ds)^2 ...