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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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Orientation of manifold’s boundary

I can’t understand the following remark from “Godinho, Natario - An introduction to Riemannian Geometry” at page 52. I know that given an orientable differential manifold $M$ with boundary, also its ...
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1answer
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Calculating the dot product of vector fields.

Given Two vector fields : $$X(x,y,z)=(x,-y,-z)$$ $$Y(x,y,z)=(1,-y,x)$$ I want to calculate the dot product of these two vector fields $X.Y$. It's just that its vector fields that is confusing me a ...
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some help with this notation.

I feel this is going to turn out to be a pretty straightforward question. I just want to make sure I'm correct in my understanding of the following notation ( It just wasn't defined in class) this is ...
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Proof of some book.

https://arxiv.org/pdf/math/0607607.pdf Theorem 5.6.(pp.110-111 ) of the above book, a quotient space are constructed for a geometric limit, but I cannot understand the construction of this quotient ...
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22 views

Tangent vector to a curve as a function

Carmo's book "Riemannian Geometry" defines a differentiable manifold and tangent vector as follows respectively. Why is the tangent vector defined as a function? Is this because in going from the ...
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1answer
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Lie group embeddings $SO(10) \supset SU(3) \times SU(2) \times U(1)?$

Let $G$ be $SO(10)$ or $Spin(10)$. Does either of them $G=SO(10)$ or $G=Spin(10)$ contain $SU(3) \times SU(2) \times U(1)$ as a subgroup? Can one show which of the following embeddings are possible ...
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Lie group embeddings $SU(5) \supset SU(3) \times SU(2) \times U(1)?$

Does the special unitary Lie group $SU(5)$ contains $SU(3) \times SU(2) \times U(1)$ as a subgroup? Can one show which of the following embeddings are possible rigorously: $$SU(5) \supset SU(3) \...
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Reparameterized geodesics - proof

I was reading the proof for the lemma that states that a certain curve c is a reparameterized geodesic if and only if it satisfies $\frac{D\dot{c}}{dt}=f(t)\dot{c}$. The starting point was defining ...
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Isn't $∇^{0,1}=\bar\partial_E+A^{0,1}$?

A connection ∇ on a holo bundle $E$ is called compatible with holo structure if $∇^{0,1}=\bar\partial_E$. And such a connection is called a Chern connection. (reference) p.17 And we know $\nabla=d+A$....
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1answer
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Proving $\left(\mathbb{R}^2_+,\frac{1}{y}(dx^2+dy^2)\right)$ is not complete

Let $\mathbb{R}^2_+:=\{(x,y)\in\mathbb{R}^2\mid y>0\}$ and the metric $g=\frac{1}{y}(dx^2+dy^2)$. Prove $(\mathbb{R}^2_+,g)$ is not complete. I guess the proper way is to find a divergent curve $\...
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Chern classes of $S^2$

It's known that $S^2$ is a $1$-dimensional complex manifold. Let $\varepsilon^n$ denote the trivial vector bundle of rank $n$, then $TS^2\oplus\varepsilon^1 = \varepsilon^3$, so by the Whitney product ...
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1answer
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Connection on the dual vector bundle

(Note: I looked at the other questions about defining a connection on the dual bundle, but the answers do not apply to my case since I use a slightly different definition of connection). I am ...
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2answers
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why do we need $4$ dimensions to embed a two dimensional shape on a surface

My Lecturer mentioned at the beginning of my differential geometry course that you need at least $4$-D to embed a $2$-D shape on an ambient space ( not too sure what ambient space means ....) My ...
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Is the group action of quotient group $G/H$ on quotient manifold $M/H$ proper?

My question arises from Chapter 21 of Lee's book, Introduction to Smooth Manifolds, 2nd edition. Let $G$ be a Lie group acting smoothly, freely and properly on a manifold $M$ on the left, denoted by ...
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Relation between Atiyah-Singer and Atiyah-Hitchin-Singer index theorems

How do we go between Atiyah-Singer index theorem to the Atiyah-Hitchin-Singer index theorem? What are the conceptual connections between the twos? And how are they reduced from one to the other?
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1answer
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Calculating arc length of a curve by pythagorean theorem

I was asked to explain the parametric curve arc length to a fellow student, only to find out that I don't completely understand it myself to be able to explain it. I've read multiple posts here about ...
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Defining a connection in $\mathbb{R}^2$ using a connection $1$-form

I'm reading Hitchin's paper Self-duality Equations on a Riemann Surface (Hitchin, self duality). In the first chapter on pages 63/64 he considers a principal $G$-bundle $P$ over $\mathbb{R}^4$ and a ...
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15 views

Can I Find ALL Connection 1-Forms from the Gauge Group?

I would like to know if its possible that given a principal fiber bundle $(P, \pi, M, G)$, the gauge group $GA(P)$ of all gauge transformations $f:P\to P$, and a single connection $1$-form $\omega_0 \...
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iWhy is a canonical Lepagean equivalent relation a functor, while classical Lepagean equivalents are not functorial in general?

enter image description here enter image description here It's a piece of Mrak Gotay's work An exterior differential system approach to the Cartan form, in that work he defined the canonical Lepagean ...
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1answer
32 views

Tetrads and metrics

I'm looking to do a similar calculation as in: Curvature tensor of 2-sphere using exterior differential forms (tetrads) in terms of tetrads/Vierbeins etc but I can't find any literature on how one ...
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Integral of Laplacian on complete non-compact manifold

Let $(M,g)$ be a complete non-compact manifold, $\Delta$ denote the Laplacian operator on smooth functions. Q Is this true that $\int_M(\Delta f\cdot f)dvol_M\geq0$ for any function $f\in L^2(M)$ ...
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1answer
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Graph of smooth functions

It's well known that if a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ ( possible multivariable) is smooth, then it's graph $((x,y) \in \mathbb{R}^{n+m}: y = f(x))$ is a smooth manifold. ...
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If ${\varphi}$ is angle between the tangent to the center of curvature of curve $c_1$ and principal normal of curve ${c}$ then [on hold]

$$tan{\varphi}=\frac{p}{(p^-){\sigma}}=\frac{(p)(T)}{p^-}$$ Where p denote the radius of carvature and T denote torsion,and $${\sigma}=\frac{1}{T}$$
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Nondegenerate symmetric covariant 2-tensors of fixed signature as an embedding.

Let $M$ be a smooth manifold of dimension $n$, and let $\Sigma^2(T^*M)$ denote the (smooth) bundle of symmetric covariant 2-tensors on $M$. If we let $\Sigma^{2,(p,q)}(T^*M)$ be the set consisting of ...
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Fibred charts adapted to principal bundle structures

If $\pi_E:E\rightarrow M$ is a rank $k$ vector bundle (let's assume everything in this question to be real for simplicity), it is the most common to use fibred charts adapted to the vector bundle as ...
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If two points of a curve have same Frenet frame and the curve's torsion and curvature are periodic then the curve is periodic

If two points $a(s_0), a(s_0+c)$ of a curve have same Frenet frame and the curve's torsion and curvature are periodic with period $c$ then the curve is periodic. I think we can assume wlog that the ...
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Is there a standard procedure for changing vector fields to different coordinate representations?

Is there a standard procedure for changing vector fields to different coordinate representations (polar, cylindrical, ...)? Regardless of what basis the field is in? I believe so, but am unable to ...
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22 views

How to prove compact regions in surfaces are closed.

This is problem 4.7.11 of O'Neill's Elementary Differential Geometry, second edition. The hint says to use the Hausdorff axiom ("Distinct points have distinct neighborhoods") and the results of fact ...
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1answer
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If $\alpha(s)$ is a simple closed regular plane curve, the tangent circular image $t$ is onto

If $\alpha(s)$ is a simple closed regular plane curve, the tangent circular image $t: [0, L] \to S^1$ is onto. Here, $\alpha$ is a curve with unit speed, and $L$ is the period. We have that, by the ...
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Matching intuition of convergence with formalism for family of spherical caps

Consider the following family of surfaces in $\mathbb R^3$. Let $P=(0,0,h)$ with $h>0$. Construct the sphere $S$ of radius $R$ centered at $(0,0,-R)$. Now draw the cone with vertex at $P$ that is ...
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Is orientability a local or global property?

That is the question in essence. The first definition of orientability is the following: A regular surface $S$ is called orientable if it is possible to cover it with a family of coordinate ...
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3answers
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Given $F(x,y)$, what does it mean to compute $dF(X)$ for $X=x \frac{d}{dx}+y\frac{d}{dy}$?

Given $F(x,y)$, what does it mean to compute $dF(X)$ for $X=x \frac{d}{dx}+y\frac{d}{dy}$? My idea: $dF(x,y)$ is the same as the Jacobian of $F(x,y)$. But in order to plug in $X$, then what should I ...
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How much area can I see on average if I’m hovering over a deformed sphere?

I’m interested in getting some estimates for how much area I can see in average if I’m above (let’s say height $h$) of a surface diffeomorphic to a sphere. Let’s start with a circle of radius $r$. ...
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1answer
31 views

Gaussian curvature of the pseudosphere

I am asked to show that the pseudosphere has Gaussian Curvature $-1$ at all points. So I parametrized the tractrix as: \begin{equation} \alpha(t) = (\sin{t},\cos{t}+\log{\tan{\frac{t}{2}}}) \end{...
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1answer
24 views

Is a B-Spline always made up of Bezier curve segments?

According to what I have read, a B-Spline curve is made up of segments, with each segment controlled by 'k' control points (where k is the order of the curve). Also, a B-Spline curve can be formed by ...
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38 views

Fundamental Neighborhood in Ordinary Differential Equations

Consider the ODE \begin{align} \dot x = f(x) \tag{1} \end{align} Let $x_0$ be a hyperbolic fixed point of $(1)$. Let $V$ be a neighborhood of $x_0$ in $W^s(x_0)$, where $W^s(x_0)$ is the stable ...
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1answer
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Is every locally Banach, Hausdorff space regular?

I am working on some infinite dimensional differential geometry. I have tried proving a somewhat weaker statement than the above by replacing locally Banach with locally metrizable. But after some ...
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Isothermal coordinates in a orientable riemannian surface

We know that any orientable riemannian surface has an isothermal coordinate around any point. This atlas provide an orientation to this surface?
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0answers
35 views

An isometry between totally geodesic submanifolds is smooth?

I am studying Sharafutdinov's Convex sets in a manifold on nonnegative curvature. I have found the following statement, $S_0$ being a totally geodesic submanifold of an open Riemannian variety $M$ ...
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22 views

$\lambda$-Lemma or Inclination Lemma in Dynamical Systems

I appreciate it if someone provide some intuitions for the $\lambda$-Lemma (Inclination Lemma) in dynamical systems? I am trying to think about a pictorial example, but I am having a hard time.
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2answers
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Is a manifold $N$ smoothly embedded in a manifold $M$ of the same dimension open in $M$?

Consider a manifold smooth manifold $N$ smoothly embedded in another manifold $M$ of the same dimension. Is it true that $N$ is open in $M$? I think this is true, due to the open mapping theorem. If ...
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Dimension of Unstable Manifolds of the Equilibrium Points Connected by a Heteroclinic Orbit

Consider the ODE \begin{align} \dot x = f(x) \tag{1} \end{align} Let $x_0$ and $y_0$ be hyperbolic critical elements (fixed points or periodic solutions) of $(1)$, and let $W^u$ and $W^s$ denote the ...
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32 views

What is the wedge product of two vector fields

I've seen the notion of a wedge product for differential forms, but I've recently come across the notation $\frac{\partial}{\partial x^i}\wedge\frac{\partial}{\partial x^j}$, where these are vector ...
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1answer
26 views

Smooth extension of a smooth map on an non-empty open subset of a manifold to the whole manifold.

Proposition: Suppose $M$ is a smooth manifold and $\emptyset\neq U\subset M$ is open and $f:U\rightarrow \mathbb{R}$ is a smooth function. Then $f$ does not necessarily extend smoothly to M. Proof: (...
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1answer
24 views

Notation in differential map between differential manifolds

When we define a differential map on $x \in M$ as an aplication between differential manifolds $f : M \to N$ such that $d(\psi \circ f \circ \phi^{-1})_{\phi(x)} : \mathbb{R}^m \to \mathbb{R}^n$ is ...
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1answer
19 views

ODE featuring product of derivatives

I consider an ODE of the form $$\frac{\mathrm{d}U(u)}{\mathrm{d}u} \frac{\mathrm{d}V(v)}{\mathrm{d}v} = f(u, v). $$ The function $f$ is known, and the goal is to find $U$ and $V$, assuming that any ...
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trace of a connection

I was wondering how to prove the following statement: Consider $E \rightarrow M$ a vector bundle. If $A \in \Gamma(\text{End}(E))$ and $\nabla: \Gamma(\text{End}(E)) \rightarrow \Gamma(\text{End}(E)\...
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$f:X \to Y$ smooth and $f(X) \subset Z$ with $Z$ submanifold of $Y$. Then $f : X \to Z$ smooth.

As written in the title, suppose that $f:X \to Y$ is a smooth function between two manifolds such that $f(X) \subset Z$ with $Z$ submanifold of $Y$. Then $f : X \to Z$ smooth regarded as function to ...
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The essential spectram of the compact manifold

I heard that the essential spectrum of the compact Riemannian manifold is empty. Why does this hold? I would appreciate if you could teach me about this.
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1answer
44 views

Definition of metric on a vector bundle

Let $\xi$ be a real vector bundle over a base space $B$. It is my understanding a metric is meant as a function, $$\beta : E(\xi \oplus \xi) \to \mathbb R$$ where $E$ denotes the total space of the ...