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 $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant.

Let $r(s)$ be a curve parametrized by arc length, and $\kappa,\kappa',\tau$ are non-zero. Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant. The teacher ...
3
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723 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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3answers
76 views

Generators of $H^1(T)$

Let $T$ denote the torus. I am working towards an understanding of de Rham cohomology. I previously worked on finding generators for $H^1(\mathbb R^2 - \{(0,0)\})$ but then realised that for better ...
2
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2answers
122 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 ...
2
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1answer
170 views

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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2answers
151 views

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 ...
2
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2answers
849 views

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 ...
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2answers
188 views

Prove that $TTM$? is a orientable bundle on $TM$

I know that $TM$ is always orientable bundle. But what is $TTM$? How try to prove that $TTM$ is a orientable bundle on $TM$.
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63 views

Poincaré Lemma problems and computing contractions in an economical way

Let $x=(A,B,C,D)$ be coordinates on $\mathbb{R}^4$. $\displaystyle \beta = \frac{(AdB-BdA)\wedge(dC \wedge dD)+(dA \wedge dB)\wedge(CdD-DdC)}{(A^2+B^2+C^2+D^2)^2}$ I would like to compute ...
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1answer
77 views

number of points of tangency of the zero divergence vector field and the equator of the sphere.

Let $V$ be vector field on the sphere $S^2$ and $\operatorname{div} V=0$. What is the minimum number tangency points of this vector field and the equator of the sphere?
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918 views

There is no isometry between a sphere and a plane.

How can I show that there is no isometry between a sphere and a plane? Wikipedia defines an isometry as follows: Let $(M,g)$ and $(M',g')$ be two Riemannian manifolds, and let $f:M\to M'$ be a ...
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1answer
1k views

What does it mean to say a boundary is $C^k$?

I need a explanation on what does it mean to say a boundary is $C^k$. Can anyone help me please. And also need some explanation on how to straighten boundary ?
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4answers
377 views

Why do we think of a vector as being the same as a differential operator?

I'm reading Frankel's The Geometry of Physics, a pretty cool book about differential geometry (at least from what I understand from the table of contents). In the first chapter, we are introduced to ...
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votes
1answer
1k views

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 $\left(\matrix{-1 & 0 \\ 0 & -2}\right)\in \text{GL}(2,\mathbb{R})$ ...
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votes
3answers
449 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 ...
6
votes
1answer
641 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 ...
6
votes
1answer
281 views

Extension of Riemannian Metric to Higher Forms

I've been reading about Riemannian manifolds, and have come across a comment that says that for a metric $g$ on an $N$-dimensional manifold $M$, considered as a bilinear map $$ g:\Omega^1(M) \times ...
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2answers
148 views

Is $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ diffeomorphic to $S^2$?

I was working on Problem 5-1 of Smooth Manifolds by Professor John Lee, and it lead me to wanting to show that $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ is diffeomorphic to $S^2$, and that is ...
5
votes
1answer
102 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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4answers
315 views

why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?

Why the following integral means the area of surface $f(x,y)=z$? $$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
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2answers
2k views

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 and the dimension of the image of f? (f being continuosly differentiable),is there a ...
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1answer
225 views

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
94 views

How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for ...
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176 views

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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2answers
346 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 ...
4
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2answers
914 views

Reconstructing space curves from its curvature and torsion

I have to write a program which gets two functions (curvature and torsion) and 3 vectors of the Frenet-Serret "frame" at the starting point - and I have to reconstruct the space curve from this given ...
4
votes
1answer
449 views

Lebesgue measure on normal matrices

Consider the space of $n\times n$ complex matrices, and equip it with its Lebesgue measure $dX$, seen as a $2n^2$-dimensional real vector space [edit: or better, a complex vector space (see the answer ...
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votes
1answer
91 views

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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votes
1answer
567 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
3
votes
1answer
175 views

If $\gamma$ is spherical, then the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Let $\gamma (t)$ be a unit-speed curve with $\kappa(t)\gt0$ and $\tau(t)\neq0$ for all $t$. Show that, if $\gamma$ is spherical, i.e., if it lies on the surface of a sphere, then ...
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2answers
144 views

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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2answers
429 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), ...
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2answers
573 views

Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
3
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1answer
381 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
1k views

planar curve if and only if torsion

Again I have a question, it's about a proof, if the torsion of a curve is zero, we have that $$ B(s) = v_0,$$ a constant vector (where $B$ is the binormal), the proof ends concluding that the ...
3
votes
1answer
563 views

Tangent space to circle

I guess I am missing something obvious here. I am reading about vector bundles. (What Karoubi calls 'Quasi Vector-Bundles') An example is the sphere, where for every point $X \in S^n$ we choose $E_X$ ...
2
votes
1answer
85 views

Sum of distances

We consider the ellipse $$\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$$ where $p>q>0$. The eccentricity of the ellipse is $\epsilon =\sqrt{1-\frac{q^2}{p^2}}$, and the points $(\pm \epsilon p, 0)$ of the ...
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2answers
69 views

Proof a $(2n-1)$-compact manifold

I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold. How give the ...
2
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0answers
163 views

Degree of odd mapping of sphere

Is it possible to prove the fact that every smooth odd mapping of $S^n$ (such that $f(x)=-f(-x)$ for every $x$) has odd degree using formula which connects degree and number of preimages of regular ...
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1answer
87 views

Hausdorff Dimension of a manifold of dimension n?

Let's say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus ...
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2answers
505 views

Divergence in spherical coordinates problem

I have this formula for the divergence of a vector field: $$\nabla_m V^m = \frac{1}{\sqrt{|g|}} \frac{\partial (V^m\sqrt{|g|})}{\partial x^m}$$ The metric tensor in spherical coordinates: $$ ...
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1answer
105 views

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 ...
2
votes
2answers
700 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 ...
2
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1answer
56 views

What are $\partial/\partial f^j$ in Jost's definition of the differential mapping?

Let $M$ be a $d$-manifold and $x_0=(x^1,x^2,\cdots, x^d)\in M$, Jost defines the tangent space at $x_0$ to be \begin{equation}\{x_0\}\times \operatorname{span}\left\{\frac{\partial}{\partial ...
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votes
2answers
1k views

Direction of the second derivative of an arclength parametrized curve

I have a question, it's so simple and stupid ._. If I have a planar curve parametrized by arc length, it's easy to show that the second derivate is orthogonal to the first derivate vector (tangent ...
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2answers
97 views

Image of any curve can be parametrized without zero derivative [closed]

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a $C^{1} ([a,b])$ injective application. Prove that there is another continuous parametrization $\rho:[c,d]\to\mathbb{R}^2$ such that the following two sets are ...
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1answer
82 views

Canonical isomorphism from $I_p/I_p^2$ to cotangent space

Sorry if the title is confusing, I don't know if the terminology is standard. For my homework this week I have to prove the following: Let $M$ be a smooth manifold and let $p \in M$. Let $I_p$ denote ...
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1answer
106 views

Product Manifold: Tangent Spaces

Problem Given a product manifold. How to prove that its tangent spaces split into direct sums: $$T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN$$ Attempts One could try the geometric perspective: ...
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1answer
84 views

zeros of a certain vector field in $\mathbb{C}P^{n}$

Let's consider the complex projective space $\mathbb{C}P^{n}$ and let $X$ be a vector field with flow given by $X_{t}:\mathbb{C}P^{n}\rightarrow\mathbb{C}P^{n}$ such that ...
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2answers
310 views

Arc length parameterization lying on a sphere

Show that if $\alpha$ is an arc length parameterization of a curve $C$ which lies on a sphere of radius $R$ about the origin then $$R^2 = ...