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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Geodesics on the torus

[This is a follow-up to my question Is there a Möbius torus?] Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five ...
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Diffeomorphisms preserving harmonic functions

I'm looking for smooth maps $ \Bbb R^n \to \Bbb R^n $ with the property that, whenever $ h $ is a harmonic function ($\Delta h=0$), $ h\circ f $ is also harmonic. Is there a nice characterization of ...
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190 views

Why spherical coordinates is not a covering?

Maybe this is an idiot question and I'm committing a trivial mistake. Let $\phi (\theta, \varphi) = (\cos \theta \sin \varphi, \sin \theta\sin \varphi, \cos \varphi)$ be the usual covering of the ...
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Is there a nowhere $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a ...
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Differential Geometry: Is a closed disk a surface?

An open disk is clearly a surface, in the sense that it is locally homeomorphic to a part of $\mathbb{R}^2$. But what about a closed disk, even though it still looks like a surface, I am starting to ...
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When does a null integral implies that a form is exact?

It is trivial to prove that the integration of a $(n-1)$ exact form on the boundary of a $n$-manifold is 0. What about the contraposative ? If the integration of a $(n-1)$-form on the boundary of a $n$...
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Differential Geometry Video Lectures

I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
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Use implicit function theorem to show $O(n)$ is a manifold

In class today our teacher mentioned that one can use the implicit function theorem to show that $O(n) \subseteq \mathbb{R}^{n^2}$ is a submanifold...that is, map $A \mapsto A^* A$, and set it equal ...
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229 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
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Real line bundle smoothly isomorphic to Möbius bundle

I am reading Lee's Introduction to Smooth Manifolds and got stuck on the problem 5.6. The question is written here, question 1. (There is a typo in the question. The last sentence should be "Show that ...
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171 views

The Affine Property of Connections on Vector Bundles

Given any two connections $\nabla_1, \nabla_2: \Omega^0 (V) \to \Omega^1 (V)$ on a vector bundle $V \to M$, their difference $\nabla_1 - \nabla_2$ is a $C^\infty (M)$-linear map $\Omega^0 (V) \to \...
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Equivalent condition for non-orientability of a manifold

I've just came across this question, which gives us a great tool for showing that smooth manifold is non-orientable. Namely Thm. If $M$ is a smooth manifold and there are two charts $(U_a,\phi_a),(...
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1answer
100 views

Can one build a homology theory using submanifolds and their boundaries?

Consider a manifold $M$, and denote by $\Delta _p M$ the set of all submanifolds of dimension $p$ (with or without boundary) of $M$. Define $G_pM$ to be the free abelian group generated by $\Delta_p ...
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How I could define a inner product in the characters in $SL(2, \mathbb R)$

I have homework in a course on Lie groups, in which I must show that $(\pi_m,\mathbb C_m[x,y])$ are the only irreducible representations of finite dimension in SL(2,ℝ). Here $ C_m[x,y])$ is the vector ...
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Proof of the second Bianchi identity

I'm asked to prove the second Bianchi identity: $$\nabla_{[e}R_{ab]c}^{\;\;\;\;d}=0$$ using the fact that: $$(\nabla_a \nabla_b -\nabla_b \nabla_a)\omega_c=R_{abc}^{\;\;\;d}\omega_d$$ For every diff. ...
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having a question on the symbol $dN_p$ when writing down its correspondence matrix

My question is about the differential of the gauss map $dN_{p}$, to start the convenient expression of the symbol and formula, I have to construct some functions(maps). First, there is a one ...
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105 views

What is the space curve with curvature and torsion obeying

$ \kappa = \cos s, \tau = \sin s $ and passing through (1,0,0), TNB triad identity matrix? previous link When numerically computed it looks like a catenoid surface of revolution for all ...
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Generating orthogonal axes on a spline

I use a Catmull-Rom spline, each section built from a set of cubic functions p = x(t), y(t), z(t). I was looking at Frenet Formulas but it's not working as desired....
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2answers
133 views

Immersion of $\mathbb{R}$ to $\mathbb{R}^2$

I have no idea how to prove that the set $\{(x, |x|): x\in\mathbb{R}\}$ is not the image of an immersion of $\mathbb{R}$ into $\mathbb{R}^2$ For example If $f(t)=(t^3, |t|^3)$ then $f'(0)=(0, 0)$.
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Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
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On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where $q:\mathbb{...
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171 views

Proving that $N$ is a manifold.

I'm dealing with the following exercise from Munkres' "Analysis on Manifolds": Let $f:\mathbb R^{n+k}\rightarrow \mathbb R^n$ be of class $C^r$. Let $M$ be the set of all $x$ such that $f(x)=0$. ...
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Non surjectivity of the exponential map to GL(2,R)

I was asked to show that the exponential map $\exp: \mathfrak{g} \mapsto G$ is not surjective by proving that the matrix $\left(\matrix{-1 & 0 \\ 0 & -2}\right)\in \text{GL}(2,\mathbb{R})$ can'...
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For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
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famous space curves in geometry history?

For an university assignment I have to visualize some curves in 3 dimensional space. Until now I've implemented Bézier, helix and conical spiral. Could you give me some advice about some famous ...
6
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720 views

A local diffeomorphism of Euclidean space that is not a diffeomorphism

Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in each point) that is not a ...
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2answers
765 views

Total Derivative and Multilinear Functions

So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review: Given a function $\mathbf{f}:\...
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3answers
515 views

Surface where number of coordinate charts in atlas has to be infinite

In the definition of a parametrised surface $S$, for every point in the surface, $p \in W \subseteq S$, where $W$ is open, there exists a coordinate chart or patch , $F :U\to \mathbb{R}^n$ that maps ...
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Jacobian matrix rank and dimension of the image

I am having problems with the following question: What is the relation between the rank of the Jacobian matrix of $f$ (which is continuously differentiable) and the dimension of the image of $f$? ...
5
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1answer
221 views

What does it mean for the group of rotation matrices to have a “manifold structure”

I have just been exposed to rotation matrices, and it is showing extremely strong connection with group theory and differential geometry which I am both totally inept at. Can someone in simple term ...
5
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1answer
111 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
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Does positive definite Hessian imply the Jacobian is injective?

Suppose $f(x):\mathbb{R}^n \mapsto \mathbb{R}$ is an infinitely differentiable function. If $\nabla^2 f(x)$, the hessian of $f$ is positive definite everywhere, does this imply that the gradient(first ...
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1answer
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Why can we think of the second fundamental form as a Hessian matrix?

Let $f: U \rightarrow \mathbb{R}^3$ be an immersion that parametrizes a piece of a surface, and let $(h_{ij})$ be the matrix for the second fundamental form of that surface. According to pg. 70 of ...
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Why do all solutions to this equation have the same form?

In this paper on page 45, the authors state that Let's assume we know that $$ w \times dw = d\varphi + \sum_{j=1}^3\alpha_jdx_j, \tag{1}$$ where $\varphi \in H^1(\mathbb{R}^3, \mathbb{R})$, $\...
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380 views

Nilpotent infinitesimals comparison

I'd like to understand better the advantages and disadvantages of various approaches to nilpotent infinitesimal numbers and their application to differential geometry in the context of physics and ...
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4answers
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Why do disks on planes grow more quickly with radius than disks on spheres?

In the book, Mr. Tompkins in Wonderland, there is written something like this: On a sphere the area within a given radius grows more slowly with the radius than on a plane. Could you explain ...
3
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1answer
420 views

Knot with genus $1$ and trivial Alexander polynomial?

I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$. A linked question could be: does there exist a Whitehead double with ...
3
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1answer
760 views

The square root of positive definite matrix

Let $M$ be the manifold of real positive definite $n \times n$ matrices, define a mapping $i:A \to \sqrt A$ (where $A\in M$ and $\sqrt A$ means the unique positive definite square root of $A$). Please ...
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1answer
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What is the definition of $dx$

I have just started to study differential forms. I don't yet fully understand the definition of what a differential form is (it's a $p$-times covariant tensor field) but I know that if $U$ is an open ...
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455 views

Parametric curve on cylinder surface

Let $r(t)=(x(t),y(t),z(t)),t\geq0$ be a parametric curve with $r(0)$ lies on cylinder surface $x^2+2y^2=C$. Let the tangent vector of $r$ is $r'(t)=\left( 2y(t)(z(t)-1), -x(t)(z(t)-1), x(t)y(t)\right)...
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Surface with non-zero mean curvature means orientable

Let $M$ be a surface in $\Bbb R^3$ with non-zero mean curvature for every point. How could I show that this implies that $M$ is orientable? By our definition, orientable means that an unitary, normal ...
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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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Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
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Prove that a curve is spherical iff it satisfies the relation

I couldn't prove that a regular curve, such that the torsion and curvature never equal zero, satisfies the relation $$\frac{\tau}{\kappa}+(\frac{1}{\tau}(\frac{1}{\kappa})´)´=0$$ iff it's spherical, i....
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What is the limit distance to the base function if offset curve is a function too?

I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
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145 views

Understanding the definition of a pullback of a differential $k$-form and applying it in $1-d$

I am having trouble understanding the definition of a pullback of a differential k-form in a basic course in differentiable geometry. This is the definition I am given. I believe it is easier to ...
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1answer
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Pullback expanded form.

Definition. If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: $$f^*\omega(x) = (df_x)^*\omega[f(x)].$$ According to Daniel Robert-...
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1answer
210 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
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842 views

Pullback of a $1$-form

All: I looked at the list of similar questions, but none seemed to be done explicitly-enough to be helpful; sorry for the repeat, but maybe seeing more examples will be helpful to many. So, I have ...
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4answers
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maximum area of a rectangle inscribed in a semi - circle with radius r.

A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum. My Try: Let ...