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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Reference for Envelope, Evolute and involute

I have to give a lecture on Envelopes, Evolute and Involute to I year undergraduate students. Please suggest me some books which explain these concepts with examples geometrically. Already I have seen ...
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contraints on equation of a cylinder

(x-a)^2+(y-b)^2 = r^2 any way to adjust this formula to add constraints to the z-axis? recently introduced to the idea this goes forever in z-axis and I want to see if theres way's to adjust formula ...
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Zeros of vectorial field [on hold]

If $M$ is a manifold in ${\mathbb R}^n$ and $X:M\rightarrow TM$ a vectorial field such that $\pi\circ X=Id$ where $\pi:TM\rightarrow M$ (projection to $M$). One zero of $X$ is such that $X(x)=(x,0)$. ...
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23 views

Flat connection with non-trivial holonomy? I cannot get it

maybe this is a dumb question, but I cannot understand how a principal $G$-bundle can have non-trivial holonomy with a flat connection. Maybe I'm missing something, but doesn't Ambrose-Singer theorem ...
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1answer
23 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
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33 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
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20 views

Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
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1answer
25 views

calculate normal field of cylinder and sphere

in a book it is given that the unit normal field of $S^2$ is $N(p)=p$ the identity map. pictorially it is clear to me. But if I take any point on the sphere and multiply(usual scalar product of ...
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1answer
18 views

How to find these “breaking” points on an offseted curve?

Check the picture: I noticed that for big offset values the offseted curves often "breaks" like this (one or more times). So my question is that what is that point marked with ':-(', and how can I ...
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45 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
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34 views

Derivative with respect to a vector and tensor on a manifold

I am reading through a paper and have come across a statement which I do not fully understand. I paraphrase below. Consider a scalar function $f = ...
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2answers
79 views

Book for Undergrad Differential Geometry

I am soon going to start learning differential geometry on my own (I'm trying to learn the math behind General Relativity before I take it next year). I got the sense that a good, standard 1st book ...
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Submanifold of $2\times 2$ Complex Matrices Using Transversality?

Let $M$ be the set of all $2\times 2$ complex matrices with $a_{21}=\bar a_{12}$. $M$ is a smooth manifold diffeomorphic to $\mathbb{R}^6$. Let $W$ be the subset consisting of all matrices in $M$ with ...
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Why is $\frac{d}{d \mu} \bigg|_{\mu=0} \, F^*_\mu F^*_\lambda t = F^*_\lambda \frac{d}{d \mu} \bigg|_{\mu=0} \, F^*_\mu t $?

In the book "Manifolds, Tensor Analysis, and Applications" by Marsden, Ratiu, Abraham the following relation (see the proof of 6.4.1, third edition) is used: $$\frac{d}{d \mu} \bigg|_{\mu=0} \, ...
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1answer
27 views

extension of a local orthonormal frame on a hypersurface

Let $N$ be a $(n+1)$-dimensional Riemannian manifold and $M\subset N$ a Riemannian hypersurface (embedded or immersed). Let $M$ and $N$ be oriented and choose a unit normal vector field $\nu$ along ...
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28 views

Why are the charts in this set all $C^\infty$-related?

I'm reading the first volume of Spivak's differential geometry series, and am having a tough time convincing myself of something mentioned in the proof of Lemma 1, Chapter 2. Let $M$ be an ...
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31 views

relative sign in hodge * of tensor product

For $\delta_i \in \bigwedge^{k_i}W_i^*$, $i=1,2$, the Hodge $*$-operator of $\delta_1 \otimes \delta_2$ is given by $$ *(\delta_1 \otimes \delta_2)=(-1)^{k_1k_2}(*_1\delta_1) \otimes(*_2 \delta_2)$$ ...
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32 views

understanding the definition of a regular surface

Let $S\subset \mathbb{R}^3$ be a subset. We call $S$ a regular surface if there exists for every point $p \in S$ an open neighbourhood $V$ of $p$ in $\mathbb{R}^3$, and if, in addition, there exists ...
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Geometrically, what is the stereographic projection of a closed $n$-ball?

To show $\overline{B^n}$ is a $n$-manifold with boundary, apparently there is a trick to use stereographic projection after subtracting out the radius connecting $0$ to the north pole. I'm familiar ...
2
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50 views

Complex structures on Riemann surfaces

Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq H^0(M;K^2)$. Considering $\alpha$ as a map $T^{0,1} M \to T^{1,0} M$, the bundle $$ \{v + \alpha(v) \mid v \in T^{0,1} M\} ...
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1answer
50 views

Submersions and induced homomorphism on fundamental groups

Suppose that $M$ and $B$ are two smooth manifolds and $\Pi:M\rightarrow B$ a submersion (and onto). Fixed $x\in M$ and $b\in B$, is the induced homomorphism $\Pi_{\#}:\pi_{1}(M,x)\rightarrow ...
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1answer
30 views

Ergodic properties of orthogonal group $O(n)$

The orthogonal group $O(n)$ is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing ...
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2answers
75 views

How is partial time derivative $\frac{\partial}{\partial t}$ defined for vector flows?

This question emerged when I was thinking about Liouville's theorem of classical mechanics. As far as I understand, the change of any function along the integral curves of Hamiltonian vector field is ...
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Vector Laplace Beltrami operator on surface tangent and surface normal vector field

Consider a closed, compact, embedded surface $f:M \rightarrow \mathbb{R}^3$ and a vectorfield $X$ on the surface that can be decomposed in the surface frame basis $\{e_1,e_2,e_3\}$, where ...
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1answer
45 views

Canonical connection on $CP^n$

I have heard something along the lines of "There is a canonical $U(1)$ connection on $CP^n$" and I am trying to understand what that means. First I suppose that the sentence refers to a line bundle ...
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12 views

A construction on principal bundles

In a paper the principal $Sp(1)$-bundle $P$ over $S^4$ is introduced as follows: let $Sp(1)\times Sp(1)\hookrightarrow Sp(2) \xrightarrow{\pi} S^4$ be the spin structure on $S^4 $. The principal ...
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2answers
66 views

The meaning of $\omega$ = df?

I'm reading Algebraic Topology by William Fulton and I am a little lost about 1-forms, which I assume to be the number of differentials according to the number of dimensions (correct me if I'm wrong. ...
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1answer
73 views

Applications of Geometry to Computer Graphics

How is differential geometry (or any type of theoretical math) related to computer graphics and/or computer programming? A friend of a friend of mine has only a bachelors degree in pure math and got ...
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2answers
41 views

Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
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Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
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47 views

Are distance-related paradoxes limited by the size of an atom?

See these 2 paradoxes: Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. ...
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1answer
14 views

Show a smooth map from a compact, connected, orientable surface to a cyllinder has singular derivative at 2 points.

Let $M$ be a compact, connected, orientable surface in $\mathbb{R}^3$. Let $N$ be the cyllinder in $\mathbb{R}^3$ defined by $x^2+y^2=1$. Suppose $f:M\to N$ is $C^{\infty}$. Show that $f_*:TM\to ...
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62 views

Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
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Can a quaternionic Kähler manifold be NOT Kähler?

I have an explicit construction of the metric on the quaternionic Kähler manifold $$\mathcal M = \frac{Sp(1, 1)}{Sp(1) \times Sp(1)}.$$ Arranging the four real degrees of freedom into two complex ones ...
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72 views

How to know if a tangent bundle is trivial from its defining equations

In this question, I am considering only regular manifolds. A Trivial Bundle The circle $S^1$ is known to have a trivial tangent bundle. As a subset of $\mathbb{R}^4$, the tangent bundle of $S^1$ ...
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59 views

Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
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Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
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1answer
18 views

Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
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1answer
42 views

The definition of scalar and vector concomitant of a metric

I'm reading Defrise-Carter's paper Conformal Groups and Conformally Equivalent Isometry Groups. One might find the paper at the following link: ...
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105 views

Hodge star operator

Again I have issues with notations. The hodge star operator is defined as : (m is the dimension of the manifold) $$\star: \Omega^{r}(M) \rightarrow \Omega^{m-r}(M)$$ $$\star(dx^{\mu_{1}} \wedge ...
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1answer
37 views

Riemannian Metric Notation

I am just being introduced to Riemannian metrics, and I am having a bit of confusion on the notation. When reading, I've encountered some different notation in different sources, so I want to make ...
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16 views

Alexandrov embedness and branch points

Let assume that $\Sigma_n$ is a sequence of compact surfaces in $\mathbb{R}^3$ of fixed genus. We assume that the surfaces are Alexandrov embedded, that is to say there exits an immersion $i_n$ from ...
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Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties $g$ is differentiable and injective $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb ...
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1answer
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Levi-Cevita symbols: Why is $\epsilon_{ijk}\epsilon_{pjk}$ equal to $2\delta_{ip}$, but not $0$?

I'm learning vector calculus on my own and sometimes strange things happen that I don't know how I should explain them. We have this famous equality: ...
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41 views

An Atlas for $\mathbb{R}/{2\pi \mathbb{Z}} $

I've been having some difficulty finding an atlas for $\mathbb{R}/{2\pi \mathbb{Z}}$. The way I have been thinking of this so far is by using the standard projection map $\pi$ on open intervals of ...
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3answers
30 views

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: ...
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1answer
38 views

Orientability of a product of smooth manifolds implies orientability of each factor

I've been learning a bit about orientability on smooth manifolds. I'm having torubles with this exercise: Given two smooth manifolds $M$ and $N$, show that the product manifold $M \times N$ is ...
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1answer
34 views

Spherical coordinates and Lebesgue integral

I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ...
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25 views

n-form density weight of the levi civita tensor

Why does the n form of levi civita $\epsilon:$=$w_{1}\wedge w_{2} ...\wedge w_{n}$ have a density of weight -1 where $w_{i}$'s are cobasis of some vector basis? I thought this geometrical quantity ...
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manifold structure on a finite dimensional real vector space

I'm reading Warner's Foundation of Differential Manifolds and Lie Groups. I don't get how the finite dimensional real vector space gives a natural manifold structure. Someone has asked it before ( ...