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Questions tagged [differential-geometry]

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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Lie algebra of vector fields on a manifold - perfect?

I feel like what I'm about to ask should be very well-known in the literature, but I just cannot find a good source, neither in my usual literature nor on stackexchange. Recall that a topological Lie ...
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32 views

Is it generally true that $\nabla\times\vec{n}=0$ for any surface or is this only true for a simply connected domain?

Is it generally true that $\nabla\times\vec{n}=0$ for any surface or is this only true for a simply connected domain? (see ftp://ftp.math.ucla.edu/pub/camreport/cam12-18.pdf) and discussion here (Curl ...
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Second isomorphism theorem for homogeneous spaces

So if a Lie group $G$ acts transitively on a manifold $X$, then $X$ is diffeomorphic to the quotient $G/H$ where $H$ is the stabilizer subgroup of a point $x$ in $X$. Now if $N$ is a normal ...
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How can a diffeomorphism be an isometry if it doesn't know about the manifolds' metric structure?

I don't understand whether an isometry in Riemannian geometry is a map between smooth manifolds, Riemannian manifolds, or neither. Wikipedia defines it as follows: Let $R = (M, g)$ and $R' = (M′, g′...
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Show that if C is a spherical helix then the projection of the helix on a plane orthogonal to its axis is an arc of an epicycloid.

This is what I have solved so far. $\frac{d(\sigma{\rho\prime})}{ds}+(\rho/\sigma)=0$(Condition for a space curve lying on sphere) s$\rightarrow$arc length of the helix For helix, $\rho$/$\sigma$=$\...
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What is the curvature form $\Omega$ associated with the Levi-Civita connection for the complexified $n$-sphere with respect to the standard metric?

What is the curvature form $\Omega$ associated with the Levi-Civita connection $\nabla^{\text{L.C.}}$ for the complexified $n$-sphere $(S^n)^{\mathbb{C}}$ with respect to the standard metric, i.e. ...
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20 views

Confused about Isometric Immersions

In do Carmo's Riemannian Geometry book, on page 125, he has the following discussion: Let $f:M^n \rightarrow \overline{M}^k$ be an immersion where $k \geq n$. Then for each $p \in M$, there exists ...
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22 views

Computing the differential of this map

Let: $G = SL(2;\mathbb{R}) = \{ (a,b,c,d) \in \mathbb{R}^4 \ \vert \ ad-bc=1 \}$, $V=\mathbb{R}^3$, and $F:G \to \mathfrak{g}^*$ defined by: $$F(a,b,c,d)=(\frac{d^2}{4}, -\frac{c^2}{4}, \frac{cd}{4} ...
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35 views

What is meant by Lee when he says “the definition of a differential was cooked up to give a coordinate independent meaning to the Jacobian matrix”

Given two smooth manifolds $M,N$ and a smooth map $F:M \to N$, it seems that the entries of the Jacobian matrix for $F$ near $a \in M$ very much should depend on the choice of coordinate charts $(\...
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Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

I found statements of the Gauss-Bonnet theorem here, here, here, here, here, here, here, and here. None of them require that the surface be orientable. However, Ted Shifrin claims in a comment to this ...
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50 views

Are Riemannian manifolds only referred to as diffeomorphic if the diffeomorphism is an isometry?

Let $M$ and $N$ be Riemannian manifolds. My understanding is that strictly speaking, a diffeomorphism $\phi:M \to N$ only acts on the smooth manifold structure, not the metric tensor. But there is a ...
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22 views

Decompose a parametric curve into monotone segments

I'm looking for a numerical computational method to decompose a parametric curve $C(t)$ by subdivision into xy-monotone segments. I searched the web thoroughly, but didn't find any usable relevant ...
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1answer
27 views

Classify the Latin alphabet in n-dimensional spaces

Consider the letters of the alphabet written as follows: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z (i) Suppose these letters are written with infinitely thin lines. Classify them according ...
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27 views

Prove that a connected manifold is path connected.

I had the following idea: to prove this by induction. Assume that we can prove it for a one-manifold. Now take a two-manifold $M$, and embed it in $R^5$. Take two points $p,q\in M$. Then there exists ...
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Why does $b(t)=\text{const}.$ follow from $<b(t),\frac{w}{|w|}\ge \text{const}.$

I want to understand the following proof. It is from the book "Differential geometry of curves and surfaces" by C. Tapp. I don't understand why $b(t)=const.$ follows from $<b(t),\frac{w}{|w|}>=...
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Area estimates for minimal surface with a given boundary

Are there any estimates for the area of surfaces spanning a given space curve $\gamma$? I would be interested in the following statements: The area of a surface spanning $\gamma$ must be larger than ...
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1answer
39 views

$\operatorname{grad}(f)$ definition and extra basis term.

Some preliminary definitions. On page 342 of Lee's Smooth Manifold he concludes that $\hat{g}(\operatorname{grad} f)(X) = Xf$ where $\hat{g}$ is the isomorphism between $TM \to T^*M$, the tangent ...
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18 views

Right invariance of a differential operator

Let $\alpha\in\mathfrak{gl}(n,\mathbb{R})$ and $F\colon GL(n,\mathbb{R})\to \mathbb{C}$ be smooth. We have the following differential operator $D_{\alpha}$ acting on $F$ by $$(D_{\alpha}F)(g)=\frac{\...
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45 views

Why does the Gauss-Bonnet theorem seem to work for some square boundary identifications but not others?

The Wikipedia article gives an interesting example of the Gauss-Bonnet theorem: As an application, a torus has Euler characteristic 0, so its total curvature must also be zero. ... It is also ...
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Can a complete surface with constant negative Gaussian curvature be isometrically embedded into $\mathbb{R}^4$?

Hilbert's theorem says that a complete surface with constant negative Gaussian curvature can't be isometrically embedded (or even immersed) into $\mathbb{R}^3$. Can it be isometrically embedded into $\...
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Differentials of non-smooth functions, wedge products of currents?

In a paper of McMullen he considers foliations on a manifold determined by a closed 1-form $\rho$. He says an $L^\infty$ function $f$ is constant on the leaves of the foliation if "$df \wedge \rho = ...
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Does knowledge of all the geodesics suffice to determine the metric up to a scaling factor?

Does knowledge of all the geodesics of a Riemannian manifold suffice to determine the metric up to a scaling factor? The metric completely characterizes the shape of a Riemannian manifold. However, ...
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Find the radius of curvature of the curve [on hold]

Find the radius of curvature of the curve: $$y^2 = \frac{(a^2)(a-x)}{x}$$ at $(a,0)$.
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+50

Solutions of a differential equation

I'm trying to solve the following differential equation and I'm stuck at what it appears to be simple calculations. I'm terribly sorry if this turns out to be really simple. $(1)$ $X(f)=2f$ where $X=...
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30 views

Under the subgroups of the group of all affine transformations what can or cannot be measured?

Edit: It was pointed out in the comments, sheers are transformations which are volume preserving, and not orthogonal. That was sloppy of me. Consider that part of my question answered. I believe ...
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2answers
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Basin of Attraction of simple nonlinear coupled ODE

Consider ($\epsilon = 0.1$) \begin{equation}\label{eq:general eq} \begin{aligned} \dot{x}_1(t) &= x_1(x_1-0.5)(x_1+0.5)+\epsilon x_2\\ \dot{x}_2(t) &= x_2(x_2-0.5)(x_2+0.5)+\epsilon x_1 \end{...
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1answer
12 views

Equation for curves running along a hemisphere of arbitrary size

Say I have a hemisphere with some radius R. Is there an equation that could represent any latitudinal line that lies along the surface of the hemisphere? I want to be able to have an equation that ...
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27 views

Intuition behind discretization of first and second fundamental form in discrete differential geometry

I'm reading this paper with the hope I understand how to discretize the first and second fundamental forms. I think the intuition behind the discretization of the first fundamental form comes from the ...
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Properties of regular curves

I’m asked the following question: Show that the following properties for a regular curve are equivalent. (a) The curve is part of a straight line (b) All its tangent lines are parallel. ...
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Quantum principal bundles in physics

Recently I was reading in Stephen B. Sontz' "Principal bundles - The quantum case" and in contrast to "the classical case" he offered almost no connections with physical concepts. For quantum groups ...
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64 views

A Hodge dual computation on a $4$-dimensional Riemannian manifold

Let $(M,g)$ be a $4$-dimensional smooth Riemannian manifold. I am trying to understand the following exterior algebra computation: Let $x^1,x^2,x^3,x^4$ be local coordinates on $M$ such that the ...
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1answer
89 views

Definition of $dz_i\otimes d\bar{z_j}(\frac{\partial}{\partial z_m},c\frac{\partial}{\partial z_n})$

Is $dz_i\otimes d\bar{z_j}(\frac{\partial}{\partial z_m},c\frac{\partial}{\partial z_n}):=dz_i(\frac{\partial}{ \partial z_m})d\bar{z_j}(\bar{c}\frac{\partial}{\partial \bar{z_n}})$? It seems that by ...
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23 views

Computing the geodesic curvature of curves in $\mathbb{H}^1 \times \mathbb{R} \subset \mathbb{L}^{3}$

Here $\mathbb{L}^3$ is $\mathbb{R}^3$ with the metric $\langle u, v\rangle_\mathbb{L} = -u_1v_1 + u_2v_2 + u_3v_3$, where $u = (u_1, u_2, u_3)$ and $v = (v_1, v_2, v_3)$ and $\mathbb{H}^1 \times \...
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proof of Poincare Lemma in Spivak's calculus on manifolds

I wonder how the equation about $I(d\omega)$ holds in Spivak's book: $I(d\omega) = \sum_{i_1<...<i_l}\sum_{j=1}^n(\int_0^1t^lD_j(\omega_{i_1,...,i_l})(tx)dt)x^j dx^{i_1}\wedge...\wedge dx^{i_l} $...
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25 views

Explicitly calculating Lie brackets with spherical coordinates

Let $\frac{\partial }{\partial z}$ be the vertical field in $\mathbb{R}^3$ and $Z\in\mathfrak{X}(\mathbb{S}^2)$ the projection of $\frac{\partial }{\partial z}$ onto the tangent space of $\mathbb{S}^2\...
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Spivak's definition of manifold: shouldn't a the set $M$ also need to satisfy the Hausdorff and second-countable properties also? [duplicate]

In the book of Differential Geometry, vol 1, by Spivak, at page 1 (to be more precise, page 1 paragraph 2) it is given that If $x \in M$, then there is some neighbourhood of the $U$ of $x$ some ...
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Submersion and non-smooth submanifolds

Suppose that we have a map $f: M \to N$ between two smooth manifolds, which is only $C^{1}$, that is differentiable with continuous differential. Let $n \in N$ be a regular value of $f$, that is, $...
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18 views

Plotting the Chen-Gackstatter Surface

As I know that the parametrization of the Chen Gackstatter surface is a complex integral. However, the integral is path-dependent, so how can I plot this surface (e.g. by choosing what paths)? I would ...
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34 views

Minimizing an integral involving first fundamental form

Suppose we have a surface parametrized by $\sigma$, (unknown) but suppose we also have parametrized surface $\sigma_0$ known I'm aiming to minimizing an integral involving a term like $$ F_I = \begin{...
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28 views

Notation of the Pullback of a $1$-form

Let $\omega = \mathrm{d}y-A(t,y)\,\mathrm{d}t$ and $\gamma(t)=(t,y(t))^t$ be given. I want to compute $\gamma^*\omega(t)=\omega_{\gamma(t)}(\dot\gamma(t))$. Now the actual problem I have, is to ...
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25 views

Do Isometries Preserve Covariant Derivatives?

O'Neill's Elementary Differential Geometry, in problem 3.4.5, asks the student to prove that isometries preserve covariant derivatives. Before solving the problem in general, I decided to work through ...
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Orientability of hypersurfaces [on hold]

I often see that when the ambient manifold is simply connected, then any closed embedded hypersurface is automatically two sided and orientable, why is that? is it a simple fact?
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1answer
32 views

Under what conditions can one glue together local diffeomorphisms?

I know that, given an open cover $\{U_\alpha\}$ of a manifold $M$ and a family of smooth maps $\,f_\alpha: U_\alpha \to N$ that agree on overlaps, it is possible to construct a smooth $f:M \to N$ that ...
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1answer
39 views

Important Background for Differential Geometry

I am a senior at a small liberal arts college, and this semester I will be taking Introduction to Differential Geometry at a big research university. I am very excited, but I have heard the course is ...
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1answer
85 views

Equations that make no sense (explaining the vector bundle isomorphism)

I'm stuck at two places in these lecture notes: 1) Consider on pp 40: What does the last equation mean? This makes no sense to me. $\varphi_i$, $i=1,2$ can't denote the $i$th coordinate function, ...
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1answer
29 views

If some of the vectors in a dual basis are orthogonal then so are the original vectors?

Let $(V,g)$ be a $2n$-dimensional real inner product space. Let $v_i$ be a basis for $V$, and let $\theta^i$ be its corresponding dual basis. ($1 \le i \le 2n$). The metric $g$ induces a metric $g^*$ ...
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1answer
33 views

Differential of scalar function is a 1-Form. What for differential of vector-valued functions?

Let $U\subset\mathbb{R}^n$ be open and $f\colon U\to\mathbb{R}$ a continuously differentiable function. The the total differential $df$ is a differential $1$-form, i.e. $df(p)$ is a cotangent vector $...
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1answer
43 views

Is $\mathbb{S}^2 \times \mathbb{R}$ homeomorphic to any subset of $\mathbb{R}^3$?

I'll be working with some $2$-dimensional surfaces in $\mathbb{S}^2 \times \mathbb{R}$ soon, and this question ocurred to me. We know that $\mathbb{S}^1 \times \mathbb{R}$ is homeomorphic to $\mathbb{...
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1answer
24 views

Is evolute unique for a space curve?

The book by T.J. Willmore states that for a space curve there are infinitely many involutes. But it emphasizes again and again that for any of the infinitely many involutes the given curve is the ...
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58 views

Riemannian holonomy of a covering

Suppose I have a connected Riemannian manifold $X$ and a covering $\pi:Y\to X$ with the pulled back metric on $Y$, making $\pi$ into a local isometry between Riemannian manifolds. Suppose we have a ...