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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Noncomplete riemannian manifolds

In Lee's Riemannian manifolds text, he claims that "on $\mathbb{R}^n$ with metric $(\sigma^{-1})^* g$ obtained from the sphere from stereographic projection, there are geodesics that escape to ...
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Lacking properties of the category of smooth manifolds

According to Wikipedia "the category of smooth manifolds with smooth maps lacks certain desirable properties"(http://en.wikipedia.org/wiki/Differentiable_manifold#Generalizations). What are these ...
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Helices in Lorentz-Minkowski space $\Bbb L^3$.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3$, with the metric: $$ds^2 = dx^2 + dy^2 - dz^2$$ (which I'll denote just by $\langle \cdot, \cdot \rangle$) If $\alpha$ is spacelike and ${\bf ...
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Question on Do Carmo

In Do Carmo's book Differential Geometry of Curves and Surfaces section 1.2 I'm trying to proof the following: Let $ \alpha: I \rightarrow \mathbb{R}^3 $ be a parametrized curve and let $ v \in ...
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134 views

Shape made by Beltrami

Beltrami made (out of thin paper or stiff cloth) a model of a surface of constant negative Gauss curvature. The original might have resembled a large saddle shaped Pringles chip, and frills might have ...
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Is the helix the unique path with constant curvature and constant torsion?

If a path has both curvature and torsion constant why it is necessarily a helix? How to prove it?
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How do I take the inner product of these two tensors: $T^{ij}$ and $T_{ij}$

The tensors are of contravariant and covariant order two, respectively. Our teacher said something about the result being identity, or the kroneker delta $\delta_i^j$, I think, but I'm not too sure. ...
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23 views

Regularity of paths

Is path $t\to(t^2,t^3)$ regular and/or piecewise regular? My understanding of regularity is that regular paths/curves derivate never vanishes $(=0)$. If I plot this curve in wolfram. It looks that ...
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37 views

Divergence in terms of Levi-Civita connection

The divergence of a vector field $X$ on a manifold $M$ is defined usually as the function $\text{Div}(.)$ such that $(\text{Div} X) \;\mu =L_X \mu$ for $\mu$ a volume form. I know that there is also ...
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47 views

Interesting and not too difficult topics in geometry

I was asked to give a talk to a mixture of undergraduate and graduate students on a topic of my choosing from differential geometry. The students will mostly not be in geometry; however, I believe I ...
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Frobenius theorem for singular foliations , what are hypotheses impose over an endomorphism $P:TM\to TM$ that it span a singular foliation in M?

Let $M$ a smooth manifold. Given a morphism $P:TM\to TM$, i.e, $\pi\circ P\equiv Id$ and is linear over the fibers. If we suppose that $P^2=P$, then for each $x\in M$, $P_{x}:T_xM\to T_xM$ is a ...
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Lie algebra and left-invariant vector fields

I want to prove that the tangent space of a Lie group at its identity $e$ is isomorphic to the vector space of left-invariant vector fields. Given an element $D \in T_e G$ (a derivation), the ...
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24 views

A smooth non-stably trivial smooth vector bundle

This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some ...
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parallel non-intersecting lines in E3

For time being I define a class of parallel lines in $ E^3 $ as lines with constant minimum distance along their common normal. Apart from helices with parametrization $ (x,y,z) = (a \cos (u) , a ...
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26 views

How to derive the standard symplectic form on a 2-sphere in cylindrical polar coordinates?

I have already know that at a point $p\in S^2$ with $u,v$ the tangent vectors at $p$. Then the standard symplectic form is $\omega_p(u,v) :=\langle p,\,u\times v\rangle$ where $u\times v$ is the outer ...
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31 views

Exact definition of a circular helix

I know that a cylindrical helix is a curve $\alpha: I \to \Bbb R^3$, such that exists a constant vector $\bf u$ which makes a constant angle with the tangent vector $\bf T$ to the curve, at every ...
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35 views

What is the circumference and area of a circle with radius r?

Let $M$ a Riemann surface with constant Gauss curvature $K = K_{0}$. Calculate the circumference and area of a circle with radius $r$. Also, calculate the geodesic curvature $K_{g}((r)$ of a circle ...
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24 views

Simple question regarding orthogonality

I'm not sure why this is tripping me up but I'm not sure what tools to use. Let $\alpha(t)$ be a parametrized curve which does not pass through the origin. If $\alpha(t_0)$ is the point of the trace ...
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43 views

Topology of a manifold

A manifold $M$ is a locally euclidian topological space (every point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$). We assume, in addition, $M$ Haussdorf and second countable. ...
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Find Area of the circle [closed]

Let C be a set in two-dimensional space and let Q(C) be the area of C if C has a finite area; otherwise, let Q(C) be undefined.Find the Q(C) if $\;C=\{(x,y)\;:\;x^2 + y^2 \le 1\}\;$.
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33 views

Calculate Brieskorn Manifold?

I need show that Brieskorn Manifold is submanifold with dimension $2n-1$ and calculate specifically for $d=2$ and $n=1$ $W(d)=\lbrace (z_{0},z_{1},...,z_{n})\in \mathbb{C}^{n+1}\vert$ $ ...
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38 views

Equivalence between orientation of the tangent bundle and orientation of manifolds

If $M^{n}$ is a manifold then the following statement are equivalent. The tangent bundle $(TM,\pi,M)$ is an orientable $n$-dimensional vector bundle. $M$ has an $\lbrace (U,h)\rbrace$ atlas on $M$ ...
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Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
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General Tensor Assistance

Sorry if this is a stupid question, but it might help me grok things if I can connect from something that's intuitive to me. Consider a transformation from Cartesian coordinates to spherical ones: ...
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42 views

Geometric Interpretation of QFT Scattering Integrals

Let $f\in C^\infty(\mathbb{R}^n,\mathbb{R}^k)$, and $g\in C^\infty(\mathbb{R}^n,\mathbb{R})$, where $k<n$. How do I compute $$\int_{\mathbb{R}^n}\delta^k(f(\mathbf{x}))\cdot ...
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Classification of Geometry

I'm asking for big picture in geometry here. I've studied the first three chapters of John Lee's smooth manifolds but still I cannot see the path ahead. My questions are mainly about classification of ...
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39 views

Is there a natural Lie bracket for $\mathfrak X (M) \times C^\infty(M)$ (pairs of vector fields and smooth functions)?

Space of smooth vector fields $\mathfrak X(M)$ on a manifold $M$ has a structure of Lie algebra with the bracket being a commutator of two vector fields. Does cartesian product $\mathfrak X (M) ...
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21 views

The normal plane to a path

PROBLEM: Let $\vec x(t)$ be a path with $\vec x'$x $\vec x'' \ne 0$ and suppose that there is a point $\vec x_0$ that lies on every normal plane to $\vec x$. Show that the image of $\vec x$ lies on a ...
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Standard complex structure in contact geometry

In symplectic geometry there is the standard model: $(\mathbb{R}^{2n}, d(\sum_{i=1}^{n}x_{i}dy_{i}))$ with standard almost complex structure $ J_{0}= \begin{pmatrix} 0 & -E_{n} \\ E_{n} & 0 ...
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1answer
64 views

Intuitive idea of immersions?

So I understand the definition of immersions and submersions, as well as the motivation for defining such ideas. Not only are they important in understanding properties of mappings of tangent spaces, ...
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43 views

Curve in an algebraic variety

Let $\lambda_1, \lambda_3, \lambda_3$ be distinct real numbers. Can it be that a curve of the form $$ t \mapsto \gamma(t) := (e^{\lambda_1 t}, e^{\lambda_2 t}, e^{\lambda_3 t}) $$ is contained for all ...
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Boothby's exercise $I.3.1$ on showing that a subset of $\mathbb{R}^{2}$ is not a manifold

First time here, so sorry for the rookie mistakes. I'm a $4^{th}$ year physics student taking Riemannian Geometry so my background on the subject is very small. I'm trying to solve Boothby's exercise ...
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2answers
38 views

What is Darboux coordinate?

What is Darboux coordinate? Is it different from coordinates from $\Bbb R^n$ or some smooth manifold? I am familiar with Riemanian manifolds, but at some how Darboux coordinates, came up in some ...
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30 views

Induced metric on the boundary of manifold and mean curvature.

This question is related to mathematics, but let me briefly explain the context: I have recently faced in physics with a Gibbons–Hawking boundary term. It's the term that we add to action when our ...
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109 views

Gradient and Laplacian in $S^1$

I'm trying to solve the particle in a ring problem without embedding the circle in $\Bbb R^3$, by instead taking the entire space to be $S^1$. Unfortunately, I haven't taken differential geometry yet ...
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64 views

What is contracting a tensor actually doing?

I'm learning about tensors, and have a vague idea regarding what contracting a tensor means—but I'm still not sure of exactly what it's doing. Maybe someone here can put it in more intuitive terms. ...
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Is $\mathbb R^n$ added by one point diffeomorphic to $S^n$?

Let $M$ be a closed smooth manifold. If for some point $p$ on $M$ we can find a diffeomorphism between $M-\{p\}$ and $\mathbb R^n$, then is $M$ diffeomorphic to $S^n$(with the standard differential ...
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Reparametrization of a curve which is not regular

Let $\alpha : [a,b] \rightarrow \mathbb R^3$ be a $C^1$ mapping (curve). Then $\alpha$ has a length. If $\alpha'(t)\neq 0$ for all $t\in [a,b]$ then, denoting $$ \sigma(t)=\int_a^t |\alpha'(u)|du, $$ ...
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Identities for differential forms and vectorfields (reference request)

Recently I found the slides of a talk of J. E. Marsden, (Differential Forms and Stokes' Theorem). These slides introduce the required objects and summarize the basics of the corresponding theory. In ...
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Derivatves of curves of hyper-sphere volumes and areas

See wikipedia "N-sphere". I need this differentiated with respect to "n", not "r". This is so I can find when the slope of the curve with respect to n(treated as x-axis), the number of dimensions, ...
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54 views

A question on Lie derivative

For a Lie derivative $\mathscr{L}_{X} Y$ of $Y$ with respect to $X$, we mean that for two smooth vector fields $X$ and $Y$ on a smooth manifold $M$ such that the following holds $$ \mathscr{L}_{X} Y = ...
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38 views

Why is the structure group for lengths $\mathbb{R}^+$ and not automorphisms in $\mathbb{R}^+$?

While reading what is a gauge to understand what a gauge is, I got stuck at a point where Terry Tao wrote: This isomorphism group is called the structure group (or gauge group) of the class of ...
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The second cohomology of total space of the $\mathbb CP^1$ bundle

$X$ is a closed smooth surface with $L$ a complex line bundle on $X$. Consider the $\mathbb CP^1$-bundle $P(L\oplus 1)$, that is the projectivization of the sum of $L$ and the trivial line bundle on ...
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Rerformulation of a previous question concerning a problem in physics that involves integration of 2-forms over the sphere

In this question the integral proposed in the posting concerns a physical problem that can shortly be described by the following : Let $J$ be a real valued function on the sphere (in fact it is a ...
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Gaussian Curvature K > 0

If M is a surface with Gaussian curvature K > 0, then the curvature of any curve C ⊂ M is everywhere positive. I was reading this in a textbook and I was trying to decide if this was true or not. I ...
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Find the cartesian equation for $(e^t,t^2)$

This isn't one I recognise. I want to express it as $f(e^t,t^2)=c$ (a level curve) but I'm not sure how. I have arrived at a partial derivative equation (knowing that in the direction $(e^t,t^2)$ ...
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Parameterise $y^2-x^2=1$ - not possible.

I'm doing stuff from a book and it has just spoke of the importance of not parameterising half a curve (with the example of a circle). However I am not sure what to do. First of all ...
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27 views

Prove that if a curves normal lines all pass through a single point, then the curve must be a circle

We can assume that the curve is parameterized by a unit speed curve, r. I know that if all of its normals are going through a point then there is some smooth function l(s) and a fixed point p such ...
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27 views

Find a rigid motion to transform the curve

Say I have a curve $$r(t)=\left(t + \sqrt3\sin t\;,\;\; 2\cos t\;,\;\; \sqrt3t-\sin t\right)$$ I have discovered it is a helix and I want to reparameterize the curve in terms of the standard helix ...
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If $\operatorname{div} X = 0$ what can be said about $X^\flat$?

If vector field $X$ is divergent free $$\operatorname{div} X = 0$$ what are the properties of a corresponding covector field $X^\flat$ (via musical isomorphism with a metric $g$)? Are there some ...