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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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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27 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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49 views

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 \...
3
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1answer
120 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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1answer
46 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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97 views

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 ...
2
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2answers
284 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 ...
3
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1answer
123 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 ...
3
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1answer
241 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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35 views

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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1answer
155 views

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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231 views

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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1answer
42 views

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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1answer
80 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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1answer
40 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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33 views

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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1answer
120 views

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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1answer
46 views

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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1answer
107 views

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 $\cosh^2-\sinh^2=...
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1answer
412 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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2answers
50 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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1answer
65 views

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 ...
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1answer
63 views

If $f$ is an immersion and $g$ is a submersion, then is $g \circ f $ a local diffeomorphism?

I don't think so; the counter example I had in mind was $f : \mathbb{R}^2 \to \mathbb{R}^3 , f(x,y) = (x,y,x)$ and $g:\mathbb{R}^3 \to \mathbb{R}^2, g(x,y,z) =(x-z, y-z)$. Is my example right?
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Push Forward on product manifold.

Some words before the question. For two smooth manifolds $M$ and $P$ It is true that $T(M\times P)\simeq TM\times TP $ If I have local coordinates $\lambda$ on $M$ and $q$ on $P$ then ($\lambda$, $q$...
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1answer
49 views

A notational issue with manifold coordinate charts.

Untidy notations in differential geometry is killing me. There are many instances of this but I really need a help to clarify this one in particular. Suppose $M$ is a manifold with a coordinate chart ...
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2answers
83 views

Christoffel's symbols for a dual connection

Suppose that $\Gamma^{\beta}_{i\alpha}$ are Christoffel symbols for a connection with respct to a (local) basis $\{E_1,...,E_n\}$. I tried to prove that the Christoffel symbols for a dual connection ...
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1answer
48 views

The differential of a smooth map on manifold at points of local maxima

I have a differentiable function $f:M \to \mathbb{R}$ where $M$ is a smooth manifold. If $p \in M $ is a point of local maxima, that is I have an open set $V \subset M$, $p \in V$, so that $f(p)\geq f(...
2
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0answers
46 views

relation between isomorphism induced by Kahler form and any other (non-degenerate) two-form

Let $V$ be an euclidean vector space, and $\omega \in \wedge^2 V^*$ be non degenerate, i.e. the induced homomorphism $\tilde\omega: V \to V^*$ is bijective. What is the relation between the two ...
2
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1answer
110 views

Parallel transport along a cardioid

I want to calculate an explicit example of a vector parallel transported along a cardioid to see what happens. Maybe someone could help me with that since no author of any book or pdf on the topic is ...
2
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1answer
265 views

How to prove it's not a manifold?

In $R^{3}$ , let $Y_{r}$ be the set of points at distance $r>0$ from the circle $C= \{ \left(x,y,z\right) ; x^2+y^2=1,z=0 \}$ i.e. a doughnut which may be too fat. How to prove that when $r \geq 1$...
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577 views

Computing the Fubini-Study metric

I would like to compute the Fubini-Study metric $g_{FS}$ for $\mathbb{CP}^{n}$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration. I tried to compute on ...
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1answer
56 views

Ruled Surface: $x(u,v) = \alpha(u) + v\beta(u)$, one of the following is true

Say we have a ruled surface that is given by $x(u,v) = \alpha(u) + v\beta(u)$ with $\alpha'$ not equal to $0$ and $\| β \| = 1$. If $\alpha'(u), \beta(u)$, and $\beta'(u)$ are linearly dependent for ...
2
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1answer
70 views

Maps between Manifolds and Maximal Rank

I'm trying to prove a theorem from Olver's Applications of Lie Groups to Differential Equations. It's supposed to be an "easy" consequence of the Implicit Function Theorem but I honestly can't see ...
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1answer
42 views

The definition of a submanifold

I am wondering why it is insufficient to define a submanifold of a manifold $M$ as a subset $S\subset M$ such that $S$ itself is a manifold. Why do we need the notions of embedded submanifolds or ...
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1answer
449 views

A curve such that all its normal lines pass through a single point

Prove that a regular curve parametrized by arc length such that all its normal lines pass through a single point is contained in a circumference. Suppose $\alpha:I\rightarrow \mathbb{R}^3$ is a ...
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111 views

geodesic polar coordinate parallel circles

When is it possible to have the same constant geodesic curvature on all parallels of a constant Gauss curvature surface? EDIT: picture added.
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2answers
55 views

n-form associated with a vector field with general metric

With the euclidean metric I use the musical isomorphisms to obtain $1$-form associated with a vector field, so for a vector field $\vec{F}=(f_1,f_2,f_3)$ we have $ \vec{F}^{\flat}=f_1dx+f_2dy+f_3dz$ ...
2
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1answer
88 views

Tangent space to $\mathbb{R}P^{n}$

I could not find any other question here related to this. If I have missed out, then this could be voted as a duplicate(Sorry if it is!). I was just trying to figure out the tangent space to the ...
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1answer
65 views

Calculating the normal vector of a surface.

Let $\alpha: I\rightarrow \mathbb{R}^3$ be a parametrized curve with non-zero curvature every where and parametrized by arc length. Let $$x(s,v)=\alpha(s)+ r(n(s)\cos v + b(s)\sin v), r\not =0, s\...
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0answers
49 views

Visualization of Gauss Bonnet geometric objects

Where can we get to see some individual surface/line combinations in isometry visualizations with constant $ \int k_g ds $ (say total tangential rotation) ? Or with constant integral curvature $\...
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1answer
335 views

Check that the parametrization x(u,v)is conformal if and only if E=G and F=0.

Check that the parametrization x(u,v)is conformal if and only if E=G and F=0. I am slightly confused with what this question is asking me. Could someone please walk me through this question. I ...
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118 views

How to parallel transport a coframe field in a geodesic normal neighborhood?

From Chern: Lectures on Differential Geometry, page 147 Chern claimed that a torsion-free connection is completely determined locally by the curvature tensor. To show that he considered a geodesic ...
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1answer
983 views

A space curve is planar if and only if its torsion is everywhere 0

Can someone please explain this proof to me. I know that a circle is planar and has nonzero constant curvature, so this must be an exception, but I am a little lost on the proof. Thanks!
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Naturality of Lie derivatives

It was left to me as an exercise to show the naturality of the Lie derivative of arbitrary tensors on a smooth manifold. Is the following argument correct (it seems too easy)? Let $X$ be a vector ...
2
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1answer
309 views

Orthogonal transformation and vector product

I found these thing in an exercise 1.5.6 in the book Differential Geometry of curves and surfaces - Do Carmo. "Show that the vector product of 2 vectors is invariant under orthogonal transformation ...
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1answer
145 views

Finding the surface area of a parametrized surface

I was wondering how you would compute the surface area of a parameterized surface. Is there a formula or set of procedures you can follow to compute this. Say I wanted to compute the surface area of a ...
0
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1answer
89 views

When to take derivative with respect to distance?

I had a previous question about the divergence in spherical coordinates and using the usual formula found on wikipedia "List of formulas in Riemannian geometry" I could not get the correct form of the ...
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0answers
71 views

Norm equivalence of p-forms

Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$. $g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$. $A^p$ denotes the set of p- forms on ...
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2answers
81 views

Implicit representations of a regular surface.

Suppose that $\mathcal{S}$ is a regular surface and $f(x,y,z)=0$ is an implicit representation of this surface in a neighbourhood $V$. Can it be shown in general that at any point of $V\cap\mathcal{S}...
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1answer
62 views

Degree of a vector field on the boundary of a punctured ball?

Suppose $v\colon \mathbb{R}^3\setminus\{x_1,\dots,x_n\}\to S^2$ is a smooth map, where all the points $x_i$ sit inside the unit ball $B^3$, and not on the boundary $S^2$. I suppose this map can be ...