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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The explicit expression for integral of forms

Could anyone please help me with the following three questions? They are simple questions, but I am confused. With a $2$-form $F=\frac{1}{2}F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$ in 4 dimension, what is ...
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If $\nabla_1$ and $\nabla_2$ are Levi-Civita connections for a metric on the smooth sphere, then their curvature tensor would recover the radius…?

I am a little confused by an idea suggested to me: putting a connection on a sphere doesn't specify a metric geometry - it remembers notions like straightness of paths, but not length of paths. Let ...
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Geodesics in upper half-plane model of $\mathbb{H}$

On this page in Schlag's book on complex analysis, he is discussing the upper half-plane model of $\mathbb{H}^2$. He says for all $z_0\in \mathbb{H}$ $$\{T'(z_0) \mid T \in PSL(2, \mathbb{R}) ...
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Holonomy computation in a sphere

Let $S^1$ be the unit sphere in $\mathbb R^3$, and let $$C=\{(r\cos t, r\sin t, h)\colon t\in \mathbb R\}$$ with $r^2+h^2=1$ be a circle in $S^2$. I want to compute the holonomy around this circle. ...
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Integration by parts on manifold with a boundary

Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and ...
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Diffeomorphism that pulls back the curvature tensor is an isometry?

I heard this statement somewhere. Can anyone provide a reference (or explanation of why this is true)? (I have also heard that the metric can be expanded as a power series in terms of the curvature ...
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Reference for theorems in Hirsch

In Hirsch's differential topology, we find the following theorems on page 31: 4.1 Theorem: Let $M$ be a $C^r$ $\partial$-manifold and $N$ a $C^r$ manifold, $r \geq 1$. Let $f : M \to N$ be a $C^r$ ...
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Poincaré hyperbolic geodesics in half-plane and disc models

EDIT1: The derivation of geodesics of the two models follows in a straightforward manner from the metric. For the half-plane we have in Cartesian coordinates $$ ds^2 = (dx^2 + dy^2)/y^2 \tag{1} ...
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Given two points in a manifold, can i find compact path-connected set that contains both

Suppose we are given two points in path-connected smooth manifold. My hypothesis is that we can find path-connected compact set that contains both. I have no idea how to prove it, in fact I don't know ...
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How to find cartesian coordinate of velocity of particle on the trajectory, $Ax^2 +2Bxy +Cy^2=1, A,B,C >0.$

Consider a particle with constant speed $|w|=w_o$ moving on trajectory $Ax^2 +2Bxy +Cy^2=1, A,B,C >0.$ Could anyone advise me how to express cartesian coordinates of $v$ in terms of $x$ and $y \ ?$ ...
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I am confused by the different definitions of manifolds.

I'm currently learning manifold from Do Carmo's Riemannian Geometry. This is his definition of differentiable manifold: But this is different from what I saw in wiki: A differentiable manifold ...
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Showing that continuous forms are zero on a $\mathscr{C}^1$ simplex $\Psi$ , if all smooth forms are zero on $\Psi$.

Question: Is the guess below correct? EDIT: There haven't been any responses yet; I wonder if the question needs to be improved somehow... Forms and simplexes are as in Rudin Rudin Principles of ...
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Applications of normal coordinates?

I am looking for applications of normal coordinates (from the exponential map) on a Riemannian manifold, since I am trying to familiarize myself with this notion. Presently the only one that I know ...
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21 views

Differential operators on compact manifolds

First I should apologise if this is a bit of a vague question, but I could not find any references for the explicit construction. I've seen it stated in several places that a differential operator on ...
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42 views

Invariance of form under flow

Take a two-dimensional symplectic submanifold $M$ in $\mathbb{R}^3$. Now, I want to show that the symplectic form $\omega$ is invariant under the Hamiltonian flow $\phi^{t}: M \rightarrow M.$ Is it ...
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$S^n$ is not a retract of the disk $D^{n+1}$ and Brouwer's Fixed Point Theorem.

I was trying to understand Hirsch's proof of this fact: "There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by ...
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how to get this matrix?

I am reading Do carmo's book and I don't understand one thing in chapter 3 about geodecics. If we define F: U$\to$M$\times$M by $F(q,v)=(q,exp_qv)$, consider the point $F(p,0)=(p,p)$, why the matrix ...
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The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ bilinear … so is it tensor like?

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ linear in both components... so is it a tensor of some kind? I know (I think) it is not a ...
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32 views

What is a $C^{n}$ surface?

I know the definition of a regular surface (a set) and a continous/differentiable/regular parametrized surface (a map). But what is a $C^{n}$ surface? Is a $C^{n}$ surface the image of a $C^{n}$ ...
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$\alpha''(t)=0$ what can we say about $\alpha$ [closed]

A parameterized curve $\alpha(t)$ has the property that its second derivative $\alpha''(t)=0,\forall t\in I$ where $I$ is some unspecified interval. What can be said about $\alpha$? I get the feeling ...
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Mistake in book on symplectic topology?

I just read the proof of the non-squeezing theorem in "Introduction to symplectic topology" by Mc Duff and Salamon. The thing that is strange is that they say: Let $\Psi$ be the linear transform ...
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Least dimension of Lie group acting transitively of a manifold

I guess that the least possible dimension of a Lie group $G$ acting smoothly and transitively on a compact manifold $M$ is $\operatorname{dim}(M)$. Is this correct and is there a ref?
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Transformation laws for tensors on general manifolds

I was interested in tensor products rather from the mathematical, abstract point of view, so topics like various tensor products in the context of, for example, Banach spaces, $C^*$-algebras and so ...
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52 views

Two definitions of conormal bundle

Suppose $X$ is a smooth manifold and $f: Z \to X$ is an immersion with transverse self-intersection. I've seen (e.g. here) the conormal bundle to $Z$ in $T^* X$ defined as Definition A: $\quad L_Z := ...
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Prove that this $1$-differential form on $S^1$ is well-defined

Let $U_i:=\{p\in S^1:x_i\ne 0\}$, $i=1,2$, be two open sets of $S^1$. Define $$ \omega_p := \begin{cases} \Bigl(\bigl(-\frac{dx_2}{x_1}\bigr)|_{U_1}\Bigr)_p\,\,\,\;\text{if}\,\,p\in U_1\\ ...
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20 views

Applying existence and uniqueness of ODE

At page no. $333$ of Spivak's Differential Geometry book , theorem $13$ says that fundamental existence and uniqueness theorem guarantees the existence of $\epsilon_1$ and $\epsilon_2 >0$ so that ...
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Zeroes of differential form embedded

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ ...
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Space of $G$-connections; respecting a spin structure

If I want to have a space of $G$-connections on a Riemann surface, I can take the fundamental group on the surface, represent its generators on $G$ and take (up to conjugation) those representation ...
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Decomposition of tangent space of principal bundle

A connection on a principal bundle $\pi:P\rightarrow M$ is a choice of horizontal subspace $H_p$ at each $p\in P$, such that $T_p P = H_p + V_p$ where $V_p = \ker((\pi_*)_p)$. It is very common to ...
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Chern class of $\mathbb C\mathbb P^2 \# 6\overline{\mathbb C\mathbb P^2}$

I would like to compute the first Chern class of $\mathbb C\mathbb P^2 \# 6\overline{\mathbb C\mathbb P^2}$ in terms of the generators of $\mathbb C\mathbb P^2$ and $\overline{\mathbb C\mathbb P^2}$. ...
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Curvature of a level set

I am using the level set method for image segmentation. In particular, the segmentation boundary $C(x, y)$ is represented as the zero level set of a level set function $\phi(x, y)$. As working on ...
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Computing curvature of hyperbolic space

Consider the unit ball in $\mathbb R^2$ endowed with the Poincaré metric: $$ds^2=\frac{4(dx^2+dy^2)}{(1-x^2-y^2)^2}.$$ I want to compute the Gaussian curvature and find that it is $-1$. Given that ...
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Chern-Gauss-Bonnet theorem for even-dimensional manifolds with boundary

On the wikipedia page for the Chern-Gauss-Bonnet theorem it states that there is a generalization of the theorem for even-dimensional manifolds with boundary, but does not provide the relevant theorem ...
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Question on symplectic geometry

I am currently reading this paper on symplectic geometry. It deals with the question how the stability properties of a sequence of (periodic) points or fixed points can be related to the second ...
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Is Hilbert's theorem generalizable to $H^3$ immersion in $\mathbb{R}^4$?

The Hilbert theorem in differential geometry concerns the immersion of the hyperbolic plane in $\mathbb{R}^3$. Is it valid for $H^3$ in $\mathbb{R}^4$?, and for all $H^n$ in $\mathbb{R}^{n+1}$?
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Question for experts in dynamical systems or symplectic geometry

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof): In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...
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Complete Riemannian metric on ${\mathbb R}^2\setminus\{0\}$.

It seems to me that the Riemannian metric $g_{ij}=\delta_{ij}/|x|^2$ on the punctured plane is complete, but I don't find a proof not involving explicit computations of the geodesic equation. Does ...
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find the tangent space of hyberboloid?

How can I find the tangent space of the hyberboloid $$ x^2 +y^2 -z^2=a$$ for $$a>0$$ in the given point: $$(\sqrt{a},0,0)$$?
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Intuitively, what is the difference between homeomorphism and diffeomorphism? Significance?

As the title suggests, intuitively, what is the difference between homeomorphism and diffeomorphism? Many thanks in advance. What is the significance of such a difference?
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Taylor series representation for a Riemannian hypersurface

This is Exercise 8.5 in Lee's Riemannian Manifolds: An Introduction to Curvature. Suppose $M\subset \mathbb R^{n+1}$ is a hypersurface with the induced metric. Let $p\in M$, and let ...
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Evaluate integral with gaussian curvature

I thought evaluating it in the following way: $$\begin{align} \int_0^{2\pi}\int_0^{\pi}K(x,y)\sqrt{\det(g_{ij})} \, dy\,dx &= \int_0^{2\pi}\int_0^\pi \sqrt{\det L_{ij}}\cdot \sqrt{{\frac{\det ...
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Definition of a complex fiber

We define a real hypersurface as a subset $M\subset\Bbb C^n$ which is locally defined as the zero-locus of some $r\in\mathcal C^2(\Omega,\Bbb R)$ ($\Omega\subseteq\Bbb C^n$ open). Then let $z_0\in M$. ...
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Implicitization of Parametric Curves

I've got a 3D parametric, smooth, simple, and closed curve given by $\sigma(s) = (\sigma_1(s),\sigma_2(s),\sigma_3(s))$ where $\sigma_1(s)$ and $\sigma_2(s)$ are given by trigonometric functions of ...
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Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$ [duplicate]

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
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Higher-Order Differential Operators as Vector Fields

On a $C^\infty$ manifold $M$, one can produce the tangent space $T_p M$ at a point by equivalence classes of tangents to smooth curves through the point $p$. When realised this way, the tangent ...
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Diffeomorphism between Euclidean space

How does one show that if $f:U\rightarrow V$ is a diffeomorphism between open sets $U\subset\mathbb{R}^m$ and $V\subset\mathbb{R}^n$ then $m=n$? Here is some working: For $u\in U$ let $v=f(u)\in V$. ...
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Normal curvature of geodesic spheres

I would like to ask the community for a reference on the following property of geodesic spheres. Let $(M,g)$ be a compact Riemannian manifold without conjugate points and $\tilde{M}$ its universal ...
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45 views

Natural operators in differential geometry?

Which operators in differential geometry is called natural? And this neutrality is respect to what property or structure? Why this is an important problem? and what is due problem relations with lie ...
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52 views

A Quotient of the Euclidean Group

$\newcommand{\euc}{\mathscr I}\newcommand{\R}{\mathbf R}$ Let $\euc(n)$ denote the the Euclidean group $\R^n\rtimes O_n(\R)$. Recall that $\euc(n)$ acts on $\R^n$ as $(\mathbf x, T)\cdot \mathbf ...
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Integrating the Hopf invariant for $\pi:S^3\to S^2$

I've been working on the last part of problem 9., chapter 9 in Nakahara's Geometry, Topology and Physics all day, with no success, and am in need of some assistance. We are asked to compute the Hopf ...