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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If 2 Equations with different variables are equal, then they are constant?

Let $\phi:\mathbb R\rightarrow\mathbb R$, $\psi:\mathbb R\rightarrow\mathbb R$ be $C^{\infty}$ maps, and $f:\mathbb R^2\rightarrow\mathbb R^3$ by $$f(u,v):(u,v,\phi(u)+\psi(v))$$ and set ...
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Isometries of a general metric

For a general (pseudo-)Riemannian manifold, i.e. in which the interval $ds$ can be written $ds^2 = g_{ab}\,dx^a \,dx^b$, is there a general prescription for finding the group of isometries- by ...
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67 views

vector field on $\mathbb{R} P^2$

Actually this is a quesion in Lee's book, Manifolds and differential geometry. I have problems working with projective spaces as manifolds.(e.g. what are curves in projective spaces ? I need to know ...
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3answers
390 views

Finding the metric of a surface embedded in $\mathbb{R}^3$

I have a problem about finding the metric of a surface defined by $x=\rho\cos\varphi,\ y=\rho\sin\varphi,\ z=\rho^2$, embedded into $\mathbb{R}^3$, where $ds^2=dx^2+dy^2+dz^2$. I have literally no ...
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Is the universal covering surface orientable?

Let $M$ be a smooth, say also closed (compact and without boundary) surface. Is it true that its universal covering surface is orientable?
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94 views

Orientation of $X \times Y$

Suppose that $X$ is not orientable. How can I show that $X \times Y$ is never orientable, no matter what manifold $Y$ may be? I've tried supposing that $X \times Y$ is orientable, then using that ...
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41 views

Differential Geometry-curves

Let $c:[0,2] \to\Bbb R^3$ be the curve given by $$c(t)=(\frac {t^3}{3},t^2,2t).$$ Then there exists an $m>0$ and a $C^{\infty}$ bijection $f:[0,m]\to [0,2]$ such that $f'(s)> 0$ for every ...
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83 views

Degree of polynomial seen as a smooth map

I need some help with a part of an exercise. Let $P$ be a real polynomial of degree $d$, seen as a map $P:\mathbb{R}\rightarrow\mathbb{R}$. Prove that if $d$ is even then the degree of $P$, $degP$, ...
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55 views

Getting started with contact bundles

I'm currently reading William Burke's book Applied Differential Geometry and he uses a lot in the development of Lagrangian Mechanics the notion of a Contact Bundle. He does explain intuitively what ...
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250 views

Relations between curvature and area of simple closed plane curves.

Let $\gamma$ be a simple closed plane curve. We know that a curve with constant curvature $\kappa$ will trace a circle in the plane. The radius of this circle is the inverse of its curvature. Now, ...
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44 views

Which integral curves of a field are defined for all times t?

Which integral curves of the field $X=x^2 \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}$ are defined for all times t? I would be very thankful if somebody can help me understand what ...
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141 views

property of a curve $\alpha(t)$

Find a parametrized curve $\alpha(t)$ whose trace is the circle $x^2 + y^2 = 1$ such that $\alpha(t)$ runs clockwise around the circle with $\alpha(0) = (0, 1)$. A parametrized curve $\alpha(t)$ ...
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71 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
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169 views

Commutator of Vector Fields

Q: Given the vector fields $A=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}$, $B=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ Calculate the commutator $\left[A,B\right]$. ...
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35 views

open connected subset of semi-Riemannian manifold

I know that an open subset of a Riemannian manifold is a Riemannian manifold. Can we say that an open connected subset of a semi- Riemannian manifold is also a semi- Riemannian manifold?
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If a connection on a principal $G$-bundle restricts to an $H$-subbundle, must its holonomy lie in $H$?

Let $P \to M$ be a principal $G$-bundle, equipped with a principal connection $D$. Let $Q \subset P$ be a principal subbundle with fiber $H$, where $H \leq G$ is a (let's say closed and connected) ...
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198 views

Question about statement of Rank Theorem in Rudin

Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open set $E\subset \mathbb{R}^n$ into $\mathbb{R}^m$, and $F'(x)$ has rank $r$ for every $x\in E$. ...
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63 views

A connection is the limit of the newton quotient of the parallel transport

Let $E\rightarrow M$ be a vector bundle with connection $\nabla$. Denote by $\Pi_{\gamma(t_{0})}^{\gamma(t_{1})}:E_{\gamma(t_{0})}\rightarrow E_{\gamma(t_{1})}$ the parallel transport map along the ...
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1answer
148 views

All points at which the surfaces $x^2+y^2+z^2-1=0$ and $x^2+y^2-z^2-2y=0$ are intersect orthogonally

$f: x^2+y^2+z^2-1=0$ $g: x^2+y^2-z^2-2y=0$ I set these two surfaces equal to each other to solve for the intersection, getting $y=(1-z^2)/2$...then attempted to insert this value of $y$ in terms of ...
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On the relation between PDEs and Distributions on Manifolds

Given a distribution $\Delta$ of dimension $n$ (continuous or smooth) in a $n+m$ dimensional manifold, can one always find a basis $\{X_j\}$ such that in local coordinates $(x^1,...x^m,y^1,...y^n)$: ...
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84 views

How to choose $f\in C^{2}(\mathbb{R})$ with compact support and takes value 1 on connected compact set?

Let $0< \delta < \pi$. My questions: (1) How to construct(choose/method) $f\in C^{2}(\mathbb R)$(= First two derivatives ($f' \ \text{and} \ f''$) of $f$ on $\mathbb R$ exists and are ...
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73 views

Lie derivative of a scalar and PDE

I posted this on the physics stackexchange, but they told me to post here, as it may be more relevant. I am reading about differential geometry, and in particular the Lie derivative and its relation ...
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$S=\{(x,y,z) \in \mathbb{R}^3: x^2+y^2=z^2, z \geq 0\}$ is not regular surface.

Suppose $S$ is a regular surface. There exists coordinate function $\textbf x:U \to S \cap V,$ for some open $U \subseteq \mathbb{R}^2$ and some open $V \subseteq \mathbb{R}^3$. WLOG, let $(0,0,0) ...
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68 views

Parameter Transformation with the Jacobian

If $\phi:U\rightarrow V$ and $\tilde{\phi}:\tilde{U}\rightarrow\tilde{V}$ are parametrizations of a regular surface $S$ with $V\cap\tilde{V}≠0$ and $V, \tilde{V}\subset S$. Let $E,F,G$ and ...
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1answer
195 views

Conformal relation for 2-dim Lorentz space-times

I have two 2-dimensional space-times ($\mathbb{S}^1\times\mathbb{R}$) with signature $(-,+)$. One of them is flat the other one has non-vanishing curvature (Riemann tensor), both have vanishing Ricci ...
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75 views

Interpretation of a line integral in complex analysis

$\newcommand{\C}{\mathbb{C}}$ Suppose $f\colon \Omega\subset \C\to\C$ is a holomorphic function and $\gamma:[0,1]\to\Omega$ is a continuous path. If $\Omega=\C\setminus\{0\}$, $\gamma(t):= e^{2\pi i ...
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78 views

Cochains: terminology

Let a real, smooth manifold $M$ be given. Let $C_k(\mathbb Z, M$) denote the set of $k$-chains with integer coefficients, and let $C_k(\mathbb R, M)$ denote the set of $k$-chains with real ...
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58 views

If $f:U\rightarrow R$ is differentiable, $U\subset R^2$, and graph(f) is a regular surface, why is?

If $f:U\rightarrow \mathbb{R}$ is differentiable, $U\subset \mathbb{R}^2$, and $\operatorname{graph}(f)=:S$ is a regular surface, why is ? $1+||\nabla f||^2=||f_x||^2||f_y||^2-2\langle ...
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150 views

Lie derivative on a riemannian manifold

Suppose we have a Riemannian manifold $(M,g,\nabla)$ with Levi-civita connection $\nabla$. We define a new symmetric non-metric connection $\bar\nabla$ on $M$. Then the Lie derivative of functions and ...
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187 views

Why 'closed differential forms' are called 'closed'?

As is well known a differential form $ \omega $ is called closed differential form if it satisfies $ \mbox{d} \omega = 0 \, \, (\ast) \, $ where $ \mbox {d} $ is the exterior derivative. I think the ...
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1answer
23 views

About $C_c^\infty((0,T)\times \Omega)$

Let $\Omega = \Omega_1 \cup \Omega_2 \cup \Gamma$ where $\Omega_1, \Omega_2$ are open domains in $\mathbb{R}^n$ and $\Gamma$ has measure zero. $\Gamma$ is the interface between $\Omega_1$ and ...
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1answer
42 views

Curve in $\mathbb{P}^{n}(\mathbb{R})$, differentiable manifolds

I need a book which contains the demonstration that if $\Gamma$ is a curve in $\mathbb{P}^{n}(\mathbb{R})$ defined as $F(x_{1}, \cdots \ \ , x_{n+1}) = 0 $ , with $F$ homogeneous polynomial, then ...
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1answer
22 views

$C^k$ hypersurfaces can be split in this way?

Let $S$ be a bounded $C^k$ hypersurface of dimension $n \geq 2$ in $\mathbb{R}^{n+1}$ with no boundary. Is it true that $S$ can be split into two hypersurfaces $S_1, S_2$ that have boundary, and a ...
3
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1answer
116 views

Vector field on sphere

I want to find the gradient vector field and flows of the function $f=x^2+2y^2+3z^2$ on the sphere $S^2$, however I've not done this in a while so would appreciate a bit of help. I'd like to see the ...
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1answer
92 views

Every principal bundle over $\mathbb{R}^n$ is trivial

On page 222 in Naber's "Topology, Geometry and Gauge fields: Foundations" there is the following remark. Using more general versions of the Homotopy Lifting Theorem one can prove that any ...
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Relation between algebraic hyper de Rham cohomology and hodge theory in positive characteristic

I have recently been looking at algebraic de Rham cohomology of curves in positive characteristic. In particular, I am looking at when the sequence $$0 \rightarrow H^0(X,\Omega_X) \rightarrow ...
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1answer
441 views

How can I parametrize Viviani's Curve?

How can I parametrize Viviani's Curve ? $\textbf{Viviani’s curve}$ is the intersection of the unit sphere with center $(\frac{-1}{2},0,0)$ and the cylinder with center $(0,0,0)$ and radius ...
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1answer
38 views

Any books on isospectral manifolds?

I was searching stuff related to M.Kac's famous question "Can one hear the shape of the drum ?" I further found results due to Gordon, Webb and Wolpert in the 2D case using Sunada method. Are there ...
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1answer
386 views

What does it mean “being geodesic” is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. ...
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77 views

Intuition for chains and cochains

I'd like to get some "geometric," "physical," (or other form of) intuition for chains, cochains, and their relationship to integration on manifolds at an elementary level. In particular, it would be ...
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148 views

Wedge product of a one form and a two form

How do we calculate $\omega \wedge \Omega$, if $\Omega$ is 2-form and $\omega = dv$? The ambient manifold is $R^{4}$ with coordinates $x$, $y$, $v$ and $w$. Thank you!
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Curvature Using Circles

Given the equation $(x - h)^2 + (y - k)^2 = r^2$ representing the family of all circles of radius r at the point $(h,k)$ if we try to form the differential equation representing this family we find an ...
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171 views

Finding the tangent plane of a point of a curve when using implicit differentiation

I need to find the tangent plane of this surface: $$(z-1)^3=\sin(y^2)e^{xz}$$ at the point $(0, \sqrt \pi ,1)$ I find $dz \over dx$ and $dz \over dy$ $${dz \over dx}={-e^{xz}\sin(y^2)z \over ...
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1answer
54 views

If differential 1-forms agree on chains with integer coefficients, are they equal?

Let $M$ be a real, smooth manifold. Let $\omega_1$ and $\omega_2$ be differential 1-forms on $M$, and let $C_1(\mathbb Z,M)$ denote the set of 1-chains with integer coefficients. If \begin{align} ...
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101 views

Consequences of tubular neighbourhood theorem

Consider an oriented manifold without boundary S embedded in an oriented manifold M. The tubular neighbourhood theorem says that there is a neighbourhood T of S in M which is diffeomorphic to ...
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1answer
119 views

Integral curves of $X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}$ on a cylinder

Consider coordinates $(\theta, z)$ on $S^1 \times \mathbb R$, and a vector field $$X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}.$$ Show that the integral curve of ...
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47 views

Show that,the curve $c:(0,2\pi)\rightarrow C$, with $c(t)=f(e^{\lambda t},t)$ intersects the cone at a constant angle.

Let $C:=\{(x,y,z)\in\mathbb R^3|z=\sqrt{x^2+y^2}\}\setminus\{0\}$ $f(u,v)=(u*cos(v),u*sin(v),u)$, where U is $U:=\{(u,v)\in R^2\ | 0<u,0<v<2\pi \}$ show that the trace of ...
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761 views

What does $dx$ mean in differential form?

This question relates to this post. From what I know in calculus and standard analysis, strictly speaking, there is no meaning of $dx$. It only makes sense when combining with another $d$, e.g. ...
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3answers
143 views

Parametrization of $y^2 - x^2=1$

I have found parametrizations for the level curve $y^2-x^2=1$, however, I have a question regarding one of them. From the Pythagorean trigonometric identity $\cos^2 x + \sin^2 x =1$ we obtain ...
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1answer
195 views

A new symmetric non-metric connection that generalizes the geodesic equation(Version 2)

A curve $\alpha$ on a riemannian manifold $(M,g,\nabla)$ is a geodesic if $\nabla_TT=0$, where $T$ is the tangent vector field. A generalization of this geodesic equation suggests that $\nabla_TT=\rho ...