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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Reparametrisation of closed not closed

I would like an example of a closed curve and a reparametrisation of the same curve that is not closed.
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21 views

$d+\Gamma$-understanding connections

$d+\Gamma$ is a formula commonly found in textbooks when one reads about connection. To be more precise: if $U$ is an open set over which a given vector bundle $E \to M$ is trivial then a coonection ...
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41 views

The curvature of surfaces in Euclidean space (Theorema Egregium)

The below animation is from Wikipedia. It shows how a helicoid can be deformed into a catanoid and vice versa without stretching. Because of this, the Theorema Egregium shows that the Gaussian ...
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17 views

Retract of a free $\Omega(\mathbb{R})$-module

Can an open subset X of $\Omega(\mathbb{R}^2)$ be an $\Omega(\mathbb{R})$-module retract of some free $\Omega(\mathbb{R})$-module? Here $\Omega(\mathbb{R}^n)$ denotes the usual topology of ...
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80 views

Pushforward along an exponential map

My differential geometry is a bit rusty, so I'd like some help with what follows: I have the following setting: $M,N$ Riemannian manifold of dimension $m<n$ with codimension $d$, $M$ in embedded ...
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$K = 0$ and $K = \text{const}$ surfaces produce $k_g = \text{constant}$ intersections?

Generally speaking, when do constant K (Gauss curvature) and zero K surfaces intersect to produce lines of constant geodesic curvature $ k_g $ ? Small circles on a sphere are examples. Or more ...
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49 views

What is the difference between abstract index notation and Ricci index notation?

I'm reading Straumann's GR text and he talks about the difference between abstract index notation and Ricci index notation very briefly. So I read the wiki article, but that did not help much. Say we ...
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23 views

On Yau's (and Schoen's) proof of the positive mass theorem

I would like to face the proof of the positive mass theorem by Yau and Schoen. I have a Bsc in Mathematics and a Msc in Theoretical Physics and I'm preparing a PhD interview-challenge where I have to ...
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1answer
44 views

Second (and higher) derivatives of maps between manifolds

I'm trying to understand derivatives of maps between manifolds, and specifically something I read in Dodson and Poston's Tensor Geometry. I'll try to provide as much background as I can for those ...
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57 views

Periodic curve on unit sphere and torsion

Define $S^2 \subset \mathbb{R^3}$ be the unit sphere. Suppose that $\alpha :\mathbb{R} \to S^2$ is a differentiable curve parametrized by arc-length. a) Show that $\kappa(s)$, the curvature of ...
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1answer
49 views

Derivative of Quadratic Form as a Linear Approximation

I'm trying to find the derivative of the $quadratic$ form, for a $symmetric$ $n$ by $n$ matrix A and $ x \in \mathbb{R}^n $, $$ f(x) = x^tAx $$ such that the derivative is a linear map from $ ...
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1answer
36 views

A differentiable map doesn't depend on the parametrization

In Do Carmo's Differential Geometry of Curves and Surfaces there's an excercise in section 2-3 that says: Prove that the definition of a differentiable map between surfaces does not depend on ...
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3answers
60 views

About the curvature of a curve

Let $\alpha : I \rightarrow R^2$ be a smooth curve. For each $t \in I$ consider $N(t)$ the normal unit vector at the point $\alpha(t)$. Fix $\lambda > 0 $ a constant and define the parallel curve ...
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2answers
72 views

Evaluating $\int_0^{2\pi} \sqrt{1+\cos(t)} \ \mathrm{d}t$

Context: I'm trying to evaluate the total length of the following curve: $\gamma: (0,2\pi) \to \mathbb{R^2}, \gamma(t)=\bigg(t+\sin(t), 1-\cos(t)\bigg)$. $$\underline{\text{My working}}$$ ...
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1answer
34 views

Derivation of the Maurer-Cartan formula

The left-invariant Maurer-Cartan forms are given by $$g^{-1}dg, $$ wher $g$ a Lie group $G$ to $M_n(\mathbb{R})$. My question is why is $$d(g^{-1}dg)=(g^{-1}dg)\wedge(g^{-1}dg)\quad ? $$ How come one ...
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1answer
21 views

Loxodrome parametric equations

I have been trying to understand HOW one arrives at the equations $x=cos(t)cos(c)$ $y=sin(t)cos(c)$ $z=−sin(c)$ of the loxodrome. I can see that if the transformation to spherical coordinates is ...
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19 views

Unit circle circumferential length on a Beltrami pseudosphere

What is the perimeter length of a geodesic circle of unit radius of tangential curvature? .. assuming that the circle lies entirely on one side of its cuspidal equator.
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3answers
50 views

How to define derivative in Minkowski space

My understanding of derivative is like this: it is the unique linear mapping that sends the difference in $x$ to the difference in $f(x)$ when the difference in $x$ is small. To put it more ...
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1answer
30 views

Is there a Poincare lemma for codifferential?

Is every co-closed form also locally co-exact? That is for each $k$-form $\omega$ such that $\delta \omega = 0$ there exists $(k-1)$-form $\eta$ for which locally $\omega = \delta \eta$. My current ...
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1answer
163 views

calculation of normal derivative

Suppose $\Omega$ is a bounded region in the plane $\mathbb{R}^2$ with smooth boundary $\partial\Omega$. Suppose $u$ is a smooth function in $\Omega$. I want to calculate ...
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3answers
47 views

Connectedness of $O(3)$ group manifold

A topological space is said to be connected if it cannot be written as $X=X_1\cup X_2$, where $X_1,X_2$ are both open and $X_1\cap X_2=\emptyset$. Otherwise, X is called disconnected. Is it wrong to ...
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1answer
68 views

Is there an easy way to reason about expressions involving lots of indices?

I have been reading some Riemannian geometry recently. So far, I think I am understanding the concepts well enough. However, I am finding it difficult to translate some of the notation into meaning. ...
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2answers
40 views

Prove this plane algeraic curve is not a differentiable manifold

Prove the algebraic curve $\{(x,y)~|~x^2(x+1)-y^2=0\}$ in $\mathbb{R}^2$ is not a differentiable manifold. Remark: It is evident that the given cubic curve has a singularity at $(0,0)$ which disable ...
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32 views

Why do Ad(K) orbits in the $-1$ eigenspace of a Cartan decomposition intersect the Weyl chamber once?

Let $G$ be a semisimple Lie group and let $\frak p\oplus t$ be a Cartan decomposition of $\frak g$ and $K$ the connected subgroup with Lie algebra $\frak t$. Choose a maximal abelian subalgebra ...
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19 views

extension of a principal connection

I am trying to prove the following: Suppose that $\alpha:H\to G$ is a Lie group homomorphism and let $P\to M$ be a principal $H$-bundle and $Q\to M$ a principal $G$-bundle. Suppose further that there ...
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1answer
21 views

Asymptotic Directions of a Cylinder

Say I am looking at a cylinder. I have found the shape operator and I have found the eigenvalues to be k1 = -1/a and k2=0. I have also found the principal directions {1,0} and {0,1}. I know that if ...
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36 views

Guessing shape made by Beltrami

If $ z = r^2 f(n \theta ) $ has constant negative Gauss Curvature, find $ f(\theta) $ or an ODE leading to it, when n is an even integer. I lost my earlier derivation involving elliptic integrals. ...
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52 views

How do I complete this proof involving the Hodge dual and inner product?

Here are the preliminaries: let $M$ be an $n$-dimensional differentiable manifold equipped with a metric $g$. Define two $p$-forms $\alpha,\beta\in \Omega^p(M)$ and an $n$-form ...
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1answer
61 views

Special reference for differential geometry

I am not entirely sure how to formulate the question, but here it is. I am looking to start a self study on general relativity, and of course I need a good grasp on semi-riemannian geometry (I am ...
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1answer
66 views

Critical Points of Quadratic Forms

just a question about finding critical points (points where the differential is not surjective). I have the equation $$ f(x) = x^tAx $$ where $A$ is a symmetric $n$ by $n$ matrix and $x$ is an element ...
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1answer
46 views

Existence of smooth diffeomorphism $f$ of open ball onto itself with $f(0) = p$.

I am trying to show that for every point $p$ of the open $n$-disk $B^n$, there exists a smooth diffeomorphism $B^n \to B^n$ sending $0$ to $p$. Certainly, it seems intuitively obvious for points $p$ ...
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105 views

Is there a codifferential for a covariant exterior derivative?

For forms on a Riemannian $n$-manifold $(M,g)$ there is a notion of a codifferential $\delta$, which is adjoint to the exterior derivative: $$\int \langle d \alpha, \beta \rangle \operatorname{vol} = ...
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1answer
58 views

Naturality of the pullback connection

I'm completely stuck proving the naturality of the pullback connection. The strategy suggested is a follows: We let $\phi: (M,g) \to (\tilde{M}, \tilde{g})$ be an isometry, with connections ...
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55 views

Determining an explicit line bundle over surface

The following is a explicitly defined complex line bundle $E\to\Sigma$ over a closed surface: View $\Sigma$ as a subset of $\mathbb{R}^3$ and consider its Gauss map $n:\Sigma\to S^2$ given by ...
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1answer
47 views

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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1answer
68 views

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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96 views

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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2answers
45 views

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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2answers
135 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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1answer
59 views

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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25 views

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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1answer
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 ...
2
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1answer
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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1answer
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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13 views

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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1answer
23 views

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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1answer
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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18 views

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 ...
2
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1answer
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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1answer
32 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 ...