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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isometric imbedding of the projective plane

How to isometrically imbed the projective plane (identifying antipodal points of the unit sphere) in $\mathbb{R^5}$? My textbook indicates that there is an isometric imbedding of the projective ...
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204 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
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205 views

Riemann tensor symmetries

The Riemann tensor has its component expression: ...
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136 views

How to define Surface Laplacian on the sphere with radius 1

The simbol $\nabla_s f$ appears in a problem of my homework, and my professor thinks it means $$\nabla_s f:= \nabla f - \hat{n}(\hat{n} \cdot \nabla f )$$ or $$ \nabla_s := (I - \hat{n}\hat{n}^T ...
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107 views

[ANSWERED]Lie brackets on vector fields

We consider $v = \frac{\partial}{\partial x}$ and $w = x * \frac{\partial}{\partial z} + \frac{\partial}{\partial y}$. I need to first find the Lie bracket between them which i get to be: ...
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113 views

Why is this a differentiable structure on the product manifold?

Suppose $M$ en $N$ are differentiable manifolds with differentiable structures $\{(U_a,x_a)\}$ and $\{(V_b,x_b)\}$ resp. Consider $M\times N$ and the mappings $z_{ab}(p,q):=(x_a(p),y_b(q))$ with $p\in ...
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93 views

Tangent map of the inclusion map of a submanifold

I'm wondering if anyone could help me with the following question let $M$ be the Minkowski spacetime, let $f\in C^{\infty}(M) ; f(m)=x^{0}(m)$, with $\{x^{\mu}\}$ being a global Cartesian coordinates ...
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143 views

Torsion and Non-metricity Tensor on a Surface

In differential geometry of surfaces, how can one define a non-zero Torsion tensor? It seems that the connection you provide has always to be symmetric since, by definition, ...
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91 views

Cohomology with Coefficients in the sheaf of distributions

It just occurred to me that one could form the sheaf of distributions $F$ on any manifold where for an open set $U$ we have $F(U)$ is the algebra of distributions on $U.$ What does cohomology with ...
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154 views

How to construct pseudospherical surfaces from sine-Gordon solutions?

Due to my not being very skilled in differential geometry, I want to ask if there is a reference (book, paper, etc.) that explicitly works out how one constructs the parametric equations of a ...
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97 views

Curvature of projected conic sections

who can prove an easy and beautiful observation on a sheet of paper in a few lines? I have used a computer algebra system to verify (the possible input is contained in the "proof") the following. Let ...
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93 views

Confusion over notation in a book on the mathematics of QFT by Faria-Melo

While formulating this question, I arrived at a likely interpretation provided in an answer to my own question below. My problem appears to be one of inexperience in working with ambient coordinates, ...
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96 views

Dimension of graphs (Differential Geometry)

I have a rather basic question about the dimension of a graph mapped by the exponential map. The problem is I can't really visualize it. It starts like that: Let $M$ be a smooth manifold of ...
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128 views

what are conormal distributions?

According to the first answer in this post, a conormal distribution $u$ on a manifold $X$ relative to a (closed, embedded) submanifold $Y$ is an element of a Banach (or Hilbert) space $H$ such that ...
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185 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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115 views

The degree of Gauss map

If $M$ is an $2m$-dimensional closed orientable hypersurface in $\mathbb R^{2m+1}$, then we have a Gauss map $G:M\rightarrow S^{2m}$. I have known from my differential geometry book that ...
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131 views

exercise on surfaces and geodesics

Maybe someone can verify my answers. The problem is as follows: Consider a line $l$ in $\mathbb{R}^3$ and rotate it around the $Oz$ axis. Denote by $A$ the set of all such obtained surfaces. ...
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50 views

Closed non-exact $2$-form on $T = S^1 \times S^1$

I would like to find a closed non-exact $2$-form on $T = S^1 \times S^1$. Here are my thoughts: Since $d\theta$ and $d\varphi$ are closed non-exact $1$-forms an obvious candidate is $d\theta \wedge ...
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29 views

Covariant derivative ambiguity

I'm studying general relativity and am running into an ambiguity with the covariant derivative. The covariant derivative acting on a scalar is, in a co-ordinate basis, simply $$\nabla_X f = X^a ...
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71 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
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14 views

Is $(-\infty,0)\times S$ for a compact closed manifold $S$ a “manifold with boundary and cylindrical ends”?

I read the following definition from this paper. Definition: Let $N$ be a Riemannian manifold with boundary $\partial N$. We say $N$ is a manifold with boundary and cylindrical ends if there ...
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49 views

What does being diffeomorphic mean in the context of configuration spaces?

A sphere space can serve as a "model space" for any configuration space that is diffeomorphic to the sphere space. This is a quote from my text book (Principles of Robot Motion: Theory, ...
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54 views

Is the tangent-cotangent isomorphism orientation preserving?

Consider $(M,g)$ a Riemannian manifold. Let's define $\varphi : TM\rightarrow T^{\ast}M$ by $\varphi(p,v):=(p,g(v,.))$, for $p\in M$ and $v\in T_{p}M$. Here, $TM$ stands for the tangent bundle and ...
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75 views

Prove that the curvature of $\gamma$ is $\frac{\kappa_{\alpha}}{\sin^2\theta}$

Let $\alpha:I\to {\mathbb R}^3$ be a cylindrical helix with a unit vector $u$ such that $u\cdot T_{\alpha}$ is a constant for all $t\in I$. For $t_0\in I$, the curve ...
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54 views

Euler characteristic of the closed unit ball

I would like to calculate the Euler-Poincaré characteristic of the closed unit ball $B$ by the de Rham cohomology, the Poincaré-Hopf theorem and Morse theory. de Rham cohomology: since $B$ is ...
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68 views

Show that $(\textbf{S}^*\textbf{B})(u,v)=\textbf{B}(\textbf{S}(u,v))\cdot \textbf{N}(u,v) \ du \wedge dv$

Let $\textbf{S}(u,v):[0,1]^2 \rightarrow \mathbb{R}^3$ be a singular $2$-cube which is smooth. Note that $0 \leq u,v \leq 1$. Let $B(\textbf{r})=B_x \ dy \wedge dz + B_y \ dz \wedge dx + B_z \ ...
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20 views

Any reparametrisation of a regular curve is regular

So I'm having a little trouble algebraically showing this is true, the hint is that it is an exercise of the chain rule. From definition, a parametrised curve $\tilde\gamma : J \rightarrow ...
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29 views

Computing the first fundamental form

Let $X$ be a smooth surface in $\mathbb{R}^{3}$. I want to compute the first fundamental form of $X$. Assume that $X$ has 2 different local parameterizations $r_1$, $r_2$ (i.e. for $r_1$ there is a ...
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30 views

no point where assumptions of Inverse Function Theorem hold?

Let $\varphi: \mathbb{R}^3 \to \mathbb{R}$, $\psi: \mathbb{R}^3 \to \mathbb{R}$ are continuously differentiable. Define $f: \mathbb{R}^3 \to \mathbb{R}^3$ by$$f({\bf x}) = (\varphi({\bf x}), \psi({\bf ...
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13 views

Out of plane cross section evolution of surfaces based on local geometry information

With this question I would like to kindly ask for feedback or general pointers to even remotely related works in regards to a challenge I face. Given a smooth surface $S$ $:\mathbb{R}^2\rightarrow ...
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37 views

Image of gauss map

Let $M$ be a compact oriented regular surface with smooth unit normal vector field $N$. Let $P =\{p\in M| K(p)\geq 0\},$ where $K$ is the Gaussian curvature. Prove that $N(P)$, the image of $P$ under ...
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56 views

Curvature of the pseudosphere

I have the parametrization $x(u,v)=(\cos u \sin v, \sin u \sin v , \cos v+\log (\tan {v/2}))$ with $0<v<\pi$ y $0<u<2\pi$. From this parametrization, how can I compute (optimally) the ...
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31 views

Inexistence of periodic orbits using Bendixson's criterion

Let $X:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a linear map. Prove that there is $\delta>0$ such that for all field $Y$ over $\mathbb{R}^2$ satisfying: $$ \underset{x\in\mathbb{R}^2}{\sup} ...
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43 views

On the Euler characteristic in Morse Theory

Let $f:M\to \Bbb{R}$ be a Morse function, where $M$ is a $k$-manifold. The index $i_{f,p}$ is defined to be the number of negative eigenvalues of the Hessian $H_f$ at the critical point $p$. For ...
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23 views

Meaning of a Hypersurface resulting from Lagrange Multipliers

Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$. We write the Lagrangian ...
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36 views

Topological structure of the Manifold valued functions

$M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is ...
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68 views

Parallel hypersurfaces in a riemannian manifold and focal points

For $M^n$ a riemannian manifold and $S$ a hypersurface, if we consider $$S_t=\{\exp^\perp(v):v\in T(S)^\perp,\;|v|=t\}$$ and $$f_t:S\rightarrow S_t:p\mapsto \exp^\perp(t\eta)$$ with $\eta$ the unit ...
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45 views

Computing the first Chern class of a manifold from the metric

I'm a physicist, and I've been tasked with computing the first Chern class of the tangent bundle of a manifold endowed with a metric, $$ds^2= f(x)^2 (dt^2+dx^2)$$ To do this, I define an orthonormal ...
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33 views

Atlas on product manifold

If {$(U_\alpha ,f_\alpha )$} and {$(V_i,y_i)$} are $C^\infty$ atlases for the manifolds $M$ and $N$ of dimensions $m$ and $n$, respectively, then the collection {$(U_\alpha \times V_i,f_\alpha \times ...
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83 views

Proving that the change of parameters is differentiable.

Let $M \subset \Bbb R^3$ be a regular surface, and ${\bf p} \in M$. Let ${\bf x} : U \subset \Bbb R^2 \to M$ and $\overline{{\bf x}}:\overline{U} \subset \Bbb R^2 \to M$ be parametrizations at ${\bf ...
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1answer
27 views

Parallel Transporting a vector

I want to parallel transport a vector $V^{\mu}$ with the initial condition $V^{\mu} = (V^{\theta},V^{\phi}) = (1,0)$ along a closed curve parameterized by $ \lambda \in [0,1]$ and determine the ...
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25 views

Equivalence of definitions of torsion of a curve

The classical definition of torsion of a curve is $\tau(s)= -B´(s)\cdot N(s)$ where B is the binormal vector and N is the normal vector but I´ve seen another definition of torsion: $\tau=lim_{\Delta ...
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How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces?

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces (explicitly)? A derivation is here but its conclusions seems not right? The expected ...
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Tangent space to lie group at identity.

I'm supposed to show that for a Lie group G, $T_{(e,e)}G\times G \simeq T_eG\oplus T_eG$ and that $T_{(e,e)}m$ is given by $(X,Y)\mapsto X+Y$. I'm having trouble proving this. I'm not exactly clear ...
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112 views

Differential-geometry textbook with solved problems

I'm looking for a textbook in differential geometry which inside has exercises with (at least) final answers. Since it's my first course in differential geometry it doesn't have to cover material (we ...
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41 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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50 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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30 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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83 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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72 views

Curvature proof of a convex plane curve

Having a little trouble with part b. Is there a way to show that this curve would be arc length paramaterized? I am assuming that we cannot say this. If it is not we can take alpha', alpha'' and ...