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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What is the kernel of a Maurer-Cartan form?

The Maurer-Cartan form on the Lie group $Gl(n,\mathbb{R})$ is a one-form taking values in $\mathfrak{gl}(n,\mathbb{R})$ as defined in the link. It has a rather concrete "extrinsic definition" as ...
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20 views

question about regular surface [closed]

Prove that set $s=\{(x,y,z):z^2+x^2+y^2=25\ \ and\ \ \ y>3\}$ is regular surface by using the definition of regular surface in book Manfredo Do Carmo
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1answer
22 views

Geodesic trajectories of 3D hyperbola

Help with a problem set question? Consider a 3-dimensional space given by the set of points $\{(x,y,z),x \in R, y \in R, z > 0\}$ with the metric $ds2 = a/z^2(dx^2 + dy^2 + dz^2)$. b) Consider ...
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1answer
91 views

Prove that the curves of the family $v^3/u^2=k$ are geodesics on a surface

Prove that the curves of the family $v^3/u^2=k$ where $k$ is a constant are geodesics on a surface with the metric $$v^2 \, du^2-2uv \, du+2u^2 \, dv^2$$ where $u,v \gt 0$.
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surface area of a Torus using differential forms,

Im studing integration over manifolds. I want to compute the surface area of the torus. Im given the usual parametrization $f(u,v)= ((a+b\cos(v))\cos(u), (a+b\cos(v))\sin(u), b\sin(v))$ for $0\leq ...
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1answer
30 views

Partial derivatives and functions equal to 0

If I have the function (family of curves) $$F(x,y,p)=(px)^2+p=0$$ I am under the impression that $$\frac{\partial F(x,y,p)}{\partial p}=2px^2+1$$ Is not always equal to $0$. Please could you ...
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1answer
260 views

Show the torus of revolution has no umbilic points

Let $\mathbb{T}$ denote the torus of revolution with the usual parametrization: $x(u,v) = ( (R + r\cos(u))\cos(v), (R + r\cos(u))\sin(v), r\sin(u) )$ Show that $\mathbb{T}$ has no umbilic ...
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30 views

Clarification on some notation and “assumptions” in page 143-144 of the book “Quantum Fields and Strings: A Course for Mathematicians, Volume 1”

I was trying to read the chapter $1$ (at page $143$) of this book Quantum Fields and Strings: A Course for Mathematicians, Volume 1 that is supposed to be an introduction to modern quantum field ...
3
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1answer
48 views

Recovering a frame field from its connection forms

I have a faced a research problem where I would need to recover a frame field given its connection forms. More precisely, I begin with an orthonormal frame field (given by data) in $\Re^3$ written as ...
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3answers
61 views

Why the name “umbilic”?

Umbilic points are points on a surface at which the principle curvatures of the surface are equal. "Umbilic(al)" refers to the navel/belly button. But why do we call these points so? What about the ...
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21 views

Integration of the components of a vector field along a curve

Let $M$ be a smooth manifold, $\gamma:[0,\tau]\rightarrow M$ a smooth curve and $X$ a vector field which does not vanish on $\gamma$ and is not tangent to $\gamma$. On $M$ we consider a vector bundle ...
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1answer
128 views

Simple exercise in cohomology

I know this is a simple exercise but I am stuck unfortunately. Question: Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
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28 views

Physical predictions using the language of manifolds?

I accept the definition of a derivative as properly motivated because it helps me make physical predictions in classical mechanics. I'm looking for something similar with manifolds. From what little ...
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1answer
24 views

Asymptotic Curves on a 1-sheeted hyperboloid

I have a 1-sheeted hyperboloid given in local coordinates by $X(u,v) = (u, v, \pm \sqrt{u^2 + v^2 - 1})$. I have to find the asymptotic curves on this surface. I have found the first fundamental ...
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1answer
44 views

Local Isometry of Sphere

How does one show that there exists no neighborhood of a point on a sphere that may be isometrically mapped into a plane? I understand that I can find the first fundamental form of the sphere $(u, v, ...
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0answers
22 views

Extension of mean-curvature normal

Suppose $M$ is a two-dimensional manifold with metric $\bar{g}$, and $r: M \to \mathbb{R}^3$ is a (not necessarily isometric) embedding of $M$ into $\mathbb{R}^3$ with first fundamental form $g$ and ...
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1answer
43 views

Notation question: What does $\langle X, - \rangle$ exactly mean?

Let $ \langle \cdot, \cdot \rangle$ be an inner product on $\mathbb{R}^n$ Then according to my course notes $$X \mapsto X^{b} = \langle X, - \rangle$$ is an isomorphism from vector fields to ...
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1answer
37 views

Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?

Given a smooth finite-dimensional vector bundle $E\to M$ on a smooth manifold $M$, is the space of smooth global sections $\Gamma(E,M)$ always a finitely generated $C^\infty(M)$ module? This amounts ...
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1answer
30 views

A literature reference for Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth Riemannian manifolds

Anyone know a respectable reliable reference for the definition of Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth compact Riemannian manifolds. It suffices for m natural and $p\geq 1$
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29 views

Notation in “Curvature in Mathematics and Physics” by Sternberg

I am not quite sure how to interpret the following statement in Sternberg's book: $$Y \times [0,h] \rightarrow R^n, (y,t) \mapsto y + t v(y)$$ Thanks
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1answer
28 views

Proving that a surface is a Möbius strip

I have a given parametrization $X(u,v)$ of a surface $S$ in $\mathbb{R}$. I must prove that it is a Möbius strip. I cannot use graphical means and I am not to reparametrize the surface- essentially, I ...
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1answer
57 views

A lemma is John Lee's Riemannian Manifold having problem with proving it

The tangential connection on an embedded submanifold $M ⊂ R^n$ is symmetric. the hint is let $X,Y$ be vector fields that are tangent of M at points of M, so is $[X,Y]$ I start with $$T(X,Y)=\nabla_x ...
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1answer
32 views

Tangent line to a differentiable curve

Let $\alpha: I \to \mathbb{R}^{n}$ be a differentiable curve such that $\alpha'(a) \neq 0$ for some $a \in I$. The line $L \subset \mathbb{R}^{n}$ through $\alpha(a)$ is the tangent line to the curve ...
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28 views

formula in differential-geometry

$(M,\omega)$ is a symplectic manifold, $\omega=d \lambda$, then I want to prove that: $$ i_v\lambda\cdot\omega^n=n\lambda\wedge i_v\omega\wedge\omega^{n-1}. $$
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31 views

Find the maximal integral curve $c(t)$ starting at the point $(a,b) \in \mathbb{R}^2$ of the given vector field.

Yet another integral curve problem. The vector field this time is $X_{(x,y)} = \dfrac{\partial}{\partial x} + x \dfrac{\partial}{\partial y}$. So, using what I learned from my last post, I should ...
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10 views

Geodesic trajectories of 3D hyperbola

Consider a 3-dimensional space given by the set of points {(x,y,z),x∈R,y∈R,z>0} with the metric ds2=a/z2(dx2+dy2+dz2). b) Consider two geodesic trajectories with initial conditions ...
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0answers
26 views

How does this integration by parts work: $\int_{Q}v\varphi_t\;dxdt = -\int_S \varphi v|_{S} \nu_t - \int_Q v_t \varphi\;dxdt$

Let $\Omega(t)$ be a bounded domain for each $t$. Let $Q=\bigcup_{t \in [0,T]} \Omega(t) \times \{t\}$ and $S=\bigcup_{t \in [0,T]} \partial\Omega(t) \times \{t\}$. The normal vector to $S$ at ...
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1answer
28 views

Explicit form of the foliation associated to a differential one-form

I'm writing because of a problem in constructiong the foliation associated to a differential one-form. Explicitely I have the following differential one-form $\theta$ on Minkowski spacetime ...
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1answer
21 views

Derivative of a vector valued function under an orthogonal transformation

Let $\alpha : \mathbb{R} \to \mathbb{R}^3$ be a space curve. I'm trying to show that its curvature, torsion, and arc length are invariant under orthogonal transformations. If $\rho: \mathbb{R}^3\to ...
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24 views

Volume form for $SE(n)$ and/or $E(n)$. [duplicate]

I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the riemannian ...
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1answer
43 views

Sobolev diffeomorphisms.

Let $M$ be a compact Riemannian manifold without boundary. Suppose $f \in H^s(M,M)$, where $H^s$ denotes the ($L^2$-based) Sobolev space. Assume $s > n/2 + 1$, so that by the Sobolev Embedding ...
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0answers
19 views

Transversality of Subbundles

It is known that transversality of submanifolds is generic in the sense that two submanifolds could be made transversal by small perturbations. I was wondering if the same is true for subbundles of ...
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27 views

Sequence of critical values has no cluster point (Milnor, Morse Theory)

The following Claim is used in the proof of Theorem 3.5 in John Milnor's "Morse Theory": Claim: Let $f: M \rightarrow \mathbb{R}$ be a differentiable function on a manifold $M$ with no degenerate ...
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58 views

Complex line bundle at symplectic manifold

Let's say that there is a symplectic manifold $(M,\omega)$ with condition of $[\omega / 2\pi ]\in H^2(M;\mathbb{Z})$. Then in what condition can I get a complex line bundle $L\twoheadrightarrow M$ in ...
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1answer
37 views

Why can't $\partial X^i/\partial x^j$ be the components of a tensor field?

From Paul Renteln, "Manifolds, Tensors and Forms" in a chapter on tensor fields: Exercise 3.22 Not every object with indices is a tensor field. Let $X = X^i \partial / \partial x^i$ be a vector ...
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1answer
30 views

Find the integral curves of the given vector field.

The vector field is as follows: $X_{(x,y)} = x \dfrac{\partial}{\partial x} - y \dfrac{\partial}{\partial y} = \begin{bmatrix} x \\y \end{bmatrix}$. I know that to find integral curves, you need to ...
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0answers
21 views

for what values of c is f a regular surface

I just want to make sure I understand what's going on here (from do carmo's differential geometry book): (a)$f(x,y,z) = (x + y + z - 1)^2$ (b)$f(x,y,z) = xyz^2$ For each function, find the critical ...
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2answers
24 views

Proof two solutions of a differential equation are linear independent

Given two solutions for a second order diferential equation: $y(x)=e^{a x}$ and $y(x)=x e^{a x}$ How to show these are linear independent? I procede as follow applying the definition of linear ...
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21 views

finding tangent planes

Question: Determine the tangent planes of $x^2+y^2-z^2=1$ at the points $(x,y,0)$ and show that they are all parallel to the $z$ axis. Proof: Let the tangent plane of $f(x,y,z)=0$ at point $(x_{0}, ...
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0answers
18 views

Explicit diffeomorphism between two models of hyperbolic 3-space

I am looking for an explicit diffeomorphism between the matrix model and the homogeneous space model of hyperbolic 3-space. Since all models are isometric, this must exist. Here, I define the matrix ...
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17 views

Explicitly reconstruct conformal mapping from conformal factor

Consider two smooth, compact surfaces $\mathcal{S_i} \subset \mathbb{R}^3$, with Riemannian metrics $g_i$, $i=1,2$ and a conformal mapping $f:S_1\rightarrow S_2$. Suppose we know the conformal ...
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25 views

Can the image of chains on a smooth manifold be thought of as a Borel $\sigma$-algebra?

Volume forms on smooth manifolds have a nice interpretation as measures, but what takes the place of the Borel $\sigma$-algebra? In particular, if we let $\mathcal{M}$ be a smooth manifold and ...
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26 views

Can we define a induced metric like this?

Let $\Sigma_r$ be a topological sphere in a 3-dimensional asymptotically flat Riemannian manifold $M$ with metric $g$, $\{\frac{\partial}{\partial x^i}\}, 1\leq i\leq3$ is the standard coordinate ...
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1answer
50 views

Why is this function not injective?

I'm working through some of Do Carmo's Differential Geometry excercises. In one of those excercises he asks if the map $f(u,v)= (u+v,u+v,uv)$ with $u>v$ is a parametrization for the plane. In the ...
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2answers
114 views

Analogue of a parabola on a sphere?

Parabola: the set of points in the plane that are equidistant from a line, called the directrix, and a point, called the focal point, not on the line. Suppose we try to replicate this on a sphere: ...
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149 views

Geodesic question

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability distributions on $\mathcal{X}=\{x_0,\dots,x_n\}$. Each $p=(p(x_0),\dots,p(x_n))\in ...
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vote
2answers
38 views

Dual space (Wikipedia)

I am struggling to understand something on Wikipedia: ''If $V$ consists of the space of geometrical vectors (arrows) in the plane, then the level curves of an element of $V^*$ form a family of ...
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38 views

Is $\nabla_{X}:\Gamma(E) \to \Gamma(E)$ continuos aplication?

Be $\Gamma(E)$ smooth sections space of an vector bundle, $\Gamma(E)$ with Fréchet topology. Gived a conexion in E and a smooth vector field $X$, Is $\nabla_{X}: \Gamma(E) \to \Gamma(E)$ continuos ...
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votes
1answer
67 views

Is this counter-intuitive result actually correct?

I was trying to calculate the surface area of part of a sphere. The result seems counter-intuitive. I have found that the surface area of a sphere increases linearly. Consider the circle, centre ...
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38 views

What is the explanation of the equation 1-4.6 in the book “Applied Exterior Calculus”?

If the $n$ function $\{f^i(x^m)\}$ are of class $C^{\infty}$ in some neighborhood of $0$, then system of autonomous ODEs $$\frac{d\bar{x}^i(t)}{dt}=f^i(\bar{x}^m(t)),\quad\quad i = 1,\cdots, n$$ has a ...