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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Difference between diffeomorphisms fixing a point or a whole neighborhood.

Let $S_g$ be a closed orientable surface of genus $g$ and $S_{g}^1$ a closed orientable surface with one boundary component. Let $p$ be in $S_g$ and let's note $\mathrm{Diff}_+(S_g,p)$ the set of ...
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1answer
135 views

Differentiability of the arc length function : $\sigma (t)=l(\alpha)[a,t]$

Let $\alpha:[a,b]\longrightarrow \mathbb{R}^n$ be a rectifiable path (not necessarily continuous) The function arc length $\sigma:[a,b]\longrightarrow \mathbb{R}$ is defined ...
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1answer
60 views

Question on Construction in Spivak's *Calculus on Manifolds*, induced transformations

First I quote the relevant passage (page 89): If we consider now a differentiable function $f : \mathbb R^n \to \mathbb R^m$ we have a linear transformation $Df(p): \mathbb R^n \to \mathbb R^m$. ...
3
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1answer
337 views

Definition of Liouville measure on energy surface of Hamiltonian system

This is a reference request, as I can't for the life of me find anything that answers my question in the literature. If $(M,\omega,H)$ is a Hamiltonian system, we know from Liouvile's theorem that ...
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2answers
153 views

Curve of length $L=1$ contained in a semicircle of diameter $2R=1$.

How prove that for any curve $\alpha(s)$ of length $L=1$ in the real plane, there is a semicircle of diameter $2R=1$ that contains it. Any hints would be appreciated.
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1answer
482 views

Advice: Algebra and category theory for geometry?

I'm interested in learning a bit of geometry. To start I'm (slowly) working my way towards differential geometry via Lee's Introduction to Smooth Manifolds. But, later on, I'd also like to study some ...
9
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1answer
139 views

Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
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1answer
63 views

The set $\{(x,y) \in \mathbb{R}^2 | x^3=y^2 \}$ is not a submanifold

Prove that the set $S=\{(x,y) \in \mathbb{R}^2 | x^3=y^2 \}$ is not a submanifold. This is the exercise from the book and I cannot understand why the chart $\phi :S \rightarrow \mathbb{R}$, ...
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2answers
77 views

differential map on holomorphic tangent space

If $f : M \to N$ is a smooth map between complex manifolds $M$,$N$ then differential map $df: T_{\mathbb{R},p} \to T_{\mathbb{R},f(p)}$ is a linear map betweeen real tangent spaces and by ...
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1answer
129 views

Transversality of Vector Fields Defined in terms of Diff. Forms and Open Books.

All: Sorry for the length of the post, but I think it is necessary to set things up so that the post is understandable. I'm trying to understand how it is that the transversality (in this case , the ...
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1answer
107 views

Doubts with differential geometry notation in Frankel

This is from Frankel's The Geometry of Physics: Problem 2.3(2) Consider the tangent bundle to a manifold $M$. Show that under a change of coordinates in $M$, $\partial/\partial q$ depends ...
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1answer
249 views

Differentiating the determinant of the Jacobian of a diffeomorphism (don't understand a proof)

For each $t$, let $A_t:\Omega_0 \to \Omega_t$ be a bi-Lipschitz map between open sets in $\mathbb{R}^n$. The map is also invertible. It satisfies $$\frac{d}{dt}A_t(y) = w(A_t(y),t)$$ where $w$ is a ...
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3answers
128 views

How to prove that $\alpha''(s)$ goes towards the inside of the curve $\alpha(s)$

Let $\alpha:[a,b]\longrightarrow \mathbb{R}^2$ be a plane curve parametrized by arc length by $\alpha(s)$. $T(s):$ unit tangent vector Note that $||T(s)||=1\Longrightarrow T'(s)\perp T(s)$ How to ...
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67 views

A question about the Euler characteristic

Let $G$ be a finite group acting freely on a compact and orientable Riemannian manifold of dimension 2. I want to show that $\chi(M/ G)=\frac{\chi(M)}{|G|}$, where $\chi$ is the Euler characteristic, ...
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77 views

Newton polygon and asymptotic behavior near a singular point

As we know, Newton polygons could be used to determine the Puiseux series of algebraic curves (see, for example, Kirwan's Complex Algebraic Curves, chapter 7). Different branches correspond to ...
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1answer
46 views

Change of Variables from Sphere to Plane

Say we are in the space $\mathbb{R}^{n}$. Consider the $n-1$ dimensional surface measure $dS$ on the boundary of the upper half sphere. We can define the coordinates on this sphere by ...
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2answers
110 views

Two definitions of integral on boundary $\int_{\partial\Omega}f$?

I have seen two definitions of an integral of a function $f:\partial\Omega \to \mathbb{R}$ from the boundary of an open set $\Omega \subset \mathbb{R}^n$ where the domain is Lipschitz. 1) ...
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1answer
73 views

Bounding an integral on boundary of Lipschitz domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with bounded boundary $\Gamma.$ So $\Gamma$ is a hypersurface of dimension $(n-1)$. I want to show that $$\int_\Gamma ...
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1answer
113 views

Reference request for differential geometry/quantum chaos text

I'm looking for a differential-geometry based exposition of chaos theory and quantum chaos. Ideally, it would start with the Hamiltonian formalism (on symplectic manifolds) and discuss as many of the ...
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2answers
207 views

Is the chart function of a smooth manifold a differomorphism, not just a homeomorphism

It's clear that a smooth chart on a manifold is a diffeomorphism. To me, the fact that smoothness of a manifold implies the smoothness of the transition function between the representation of two ...
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174 views

Definition of dual connection

How to solve the following: Let $\pi:E\rightarrow M$ be a vector bundle with connection $\nabla$ and $\pi:E^*\rightarrow M$ dual bundle (such that ${\pi^*}^{-1}(p)$ is a dual of $\pi^{-1}(p)$). How ...
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0answers
66 views

Inverse Function Theorem (results using it)

Hi i'm thinking in some ideas for my bachelor thesis. I'm working in a more "general" framework than manifolds, and i found that the Inverse Function Theorem is valid in such structures. So i was ...
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1answer
2k views

Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
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179 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
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1answer
81 views

A question about covariant derivative (find $D_{v_p}W$)

the quest. on my book is $(y_1,y_2,y_3)\quad R^3\quad coordinate \quad system$ . if $W=y_1y_2^2\frac{\partial}{\partial y_1}+(y_3-y_2^2)\frac{\partial}{\partial y_2}+3y_1\frac{\partial}{\partial ...
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116 views

The relation between conformally related metrics and conformal vector fields?

Two metrics $g_{1}$ and $g_{2}$ are conformally equivalent metrics if $g_{2}=e^{2\theta}g_{1}$ A vector field $X$ is called conformal if $L_{X}g=2\theta g$ where $L_{X}$ is the Lie derivative with ...
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1answer
184 views

Relation of Hodge Theorem to Eigenfunction Basis of Laplacian

The classical Hodge theorem I know of relates the de Rham cohomology groups isomorphically to the space of harmonic forms and shows that $Id=\pi+\Delta G$, where $\pi$ is the harmonic projection of ...
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34 views

What can we say about the integral curve of a vector field on the warped product manifold?

Let $Z=(X,Y)$ be a vector field defined on the warped product $M×_{f}N$ where $f$ is defined on $M$. The integral curve of $X$ on $M$ is $\alpha$ and the integral curve of $Y$ on $N$ is $\beta$. I ...
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3answers
191 views

checking the diffeomorphism between 2 surfaces

I tried to show that a surface $x^4+y^2+z^2=1$ and the unit sphere are diffeomorphic. Since a diffeomorphism between them is not chosen easily, I would apply a theorem using invertibility of ...
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1answer
42 views

How prove this $\frac{dr_{1}}{dr_{2}}=\frac{\cos{\theta_{1}}}{\cos{\theta_{2}}}$

Question: let two point $O_{1},O_{2}$, such $$PO_{1}=r_{1},PO_{2}=r_{2}$$ There is a curve $AB$ and the point $P\in AB$,and EF is a tangent curve AB,and the tangent is $P$. let $$\angle ...
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1answer
169 views

Integral of Square of Mean Curvature

(1)If $T^{2}$ represents standardized torus and $H$ is its mean curvature, then wen have $\int_{T^{2}}H^{2}dV\geq2\pi^{2}$. (2)If $\Sigma$ represents a compact surface whose $g=1$ in $E^{3}$ and $H$ ...
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4answers
504 views

Reason for Continuous Spectrum of Laplacian

For the circle $S^1$, it is well-known that the Laplace-Beltrami operator $\Delta=\text{ div grad}$ has a discrete spectrum consisting of the eigenvalues $n^2,n\in \mathbb{Z}$, as can be seen from the ...
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95 views

Geometric Cauchy Problem

I'm attending a course in Symplectic Mechanics and I have some problems in understanding something written in my lecture notes. We are in the following setting: let $Q$ be a manifold (of dimension ...
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2answers
46 views

how to show $\frac{\partial\hat\sigma}{\partial\hat u}\times\frac{\partial\hat\sigma}{\partial\hat v}$ (cross product)

how to show $$\frac{\mathcal{\partial\hat\sigma}}{\partial\hat u}\times\frac{\mathcal{\partial\hat\sigma}}{\partial\hat v}=\left(\frac{\mathcal{\partial u}}{\partial\hat u} \frac{\mathcal{\partial ...
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1answer
163 views

Lie derivative is zero $\Leftrightarrow$ “$\phi_t$ is a symmetry transformation for $T$ $\,\,\forall t\in\mathbb{R}$.”

Let $T$ be a smooth tensor field on a manifold $\mathcal{M}$. Let $\phi\in\mathcal{M}^\mathcal{M}$ be a diffeomorphism. Question 1: I'm wondering if "$\phi$ is a symmetry transformation for $T$" iff ...
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1answer
23 views

Finding differential from 3 sampled points

I have 3 equidistant sampled points with values (-1,ym1), (0,y0), (1,y1). I would like to find an exact differential at point (0,y0). Is this doable at all, or should I evaluate the whole sampled data ...
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1answer
245 views

Calculate normal vector to $2$-face of polytope in $\Bbb R^n$

I am trying to work through a divergence theorem application for a function integrated over an $n$-dimensional convex polytope, but I can't seem to figure out how to properly calculate the normal ...
3
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2answers
262 views

Should diffeomorphisms preserving arc length be affine?

Problem Suppose $\varphi\colon V=\mathbb R^n\to V$ be a differmorphism and $d\varphi$ is its tangent mapping. $\langle\circ,\circ\rangle$ is a nondegenerate (symmetric or symplectic) bilinear form on ...
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2answers
113 views

differmorphism and homeomorphism for manifolds

For two abstract manifolds that are differmorphic, why are they always homeomorphic? Why does differentiability imply continuity for abstract manifolds? (for $R^n$ this is certainly clear)
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1answer
108 views

Is there any way to define differentiablity without any reference to the Euclidean space?

We define metric spaces based on the properties of the real numbers $\Bbb{R}$. In the same spirit we define smooth manifolds. But there is a more general and elegant way to formulate our intuition of ...
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0answers
50 views

Most natural symplectic structure?

Suppose I have 2-dimensional manifold embedded in $\mathbb{R}^3$. It's clear that the most natural Riemannian metric is the one induced by the usual inner product. What about symplectic forms? Is ...
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2answers
108 views

transition functions of a vector bundle

why are the transition functions in the definition of a vector bundle $P:E \rightarrow M$ termed as transition functions? they are $g_{\alpha \beta}:U_{\alpha} \cap U_\beta \rightarrow ...
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2answers
157 views

when can you estimate curvature from finite information about two geodesics?

Let $c_v, c_w$ be two geodesics starting at a point $p\in M$, where M is a nonpositively curved, complete, smooth Riemannian manifold. Say $c_v(\varepsilon) = \exp_p(\varepsilon v)$ and ...
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2answers
315 views

On the definition of a geodesic in a metric space

I am interested in the definition of geodesics in metric spaces. A definition which seems reasonable to me is that a geodesic should locally be a distance minimizer. Wikipedia ...
3
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1answer
273 views

What is the importance of conformal vector fields on Riemannian manifolds?

A vector $X$ on a Riemannian manifold $(M,g)$ is called conformal if $L_{X}(g)=2sg$ where $L_{x}$ is the Lie derivative and $s$ is a real-valued function on $M$. If $s=0$, $X$ is called a killing ...
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0answers
153 views

A formula for the holomorphic sectional curvature.

I tried to compute the holomorphic sectional curvature of a hypersurface of ($\mathbb{C}^{n+1}$, std metric, i), but I failed. $$ V_{k}=\{(z_{0},...,z_{n})\in \mathbb{C}^{n+1} | \sum_{j}z_{j}^{k}=0\}- ...
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1answer
129 views

open sets in a regular surface in $R^3$

I am reading Differential geometry of curves and surfaces by Do Carmo. Let S be a regular surface in $R^3$. I wonder How is open set of S defined? Is a subset of S open if and only if it is the ...
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1answer
103 views

Genus 2 surface equipped with hyperbolic metric is not a symmetric space.

Because the genus 2 surface with hyperbolic metric has constant sectional curvature, its Riemann tensor is covariantly zero. i.e. $\bigtriangledown R \equiv 0$ Therefore, it is locally symmetric ...
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2answers
1k views

Length of chord on ellipse

Suppose I have an ellipse centered at the origin, preferably expressed in its matrix form, and I want to know the chord length of a segment that passes through the origin with the endpoints at the ...
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1answer
132 views

holomorhicity implies harmonic function in several variables

I had read somewhere that it follows by cauchy riemann equations that any holomorphic or anti-holomorphic function $f$ from an open subset of $C^n$ to $C$ is harmonic i.e $\sum_{i=1}^n ...