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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Transform Confocal Ellipsodal to Spherical Coordinates

I am having trouble transforming confocal ellipsoidal coordinates to spherical coordinates. How does one perform such a transform?
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Simplify Laplace equation in rectangle geometry

Consider Laplace's equation in a rectangle as shown in the following figure. The boundary conditions are shown in the figure. The problem is solved in the case of a1 =a2=1. Is there a way to ...
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19 views

An explicit Lorentzian metric on the Klein bottle

I want to construct an explicit Lorentzian metric on the (abstract) Klein bottle but have no idea where to start. Could someone please give me a hint and/or guide me in the right direction? Thanks.
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33 views

Left-Invariant Vector Fields: Smoothness

Given a Lie group. Rough left-invariant vector fields are smooth: $$X_g:=dl_gv:\quad X\in\mathcal{X}(M)$$ How to prove this in a clever way? (I've seen some more or less technical proofs.)
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1answer
14 views

Understanding Lipschitz domain

Here is the definition of Lipschitz domain given by Wikipedia. Let n ∈ N, and let Ω be an open subset of Rn. Let ∂Ω denote the boundary of Ω. Then Ω is said to have Lipschitz boundary, and is called ...
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1answer
25 views

Finding the unit normal vector

Q. Consider the following vector function. $$ r(t)= \langle 6\sqrt{2}t,e^{6t},e^{-6t} \rangle $$ Find the unit tangent and unit normal vectors T(t) and N(t). I found $$T(t)= ...
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1answer
22 views

Product Manifold: Tangent Spaces

Problem Given a product manifold. How to prove that its tangent spaces split into direct sums: $$T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN$$ Attempts One could try the geometric perspective: ...
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37 views

Derivations: Characterization

Given a smooth manifold. (In fact, it seems irrelevant to regard manifolds.) Regard germs of functions: $$\mathcal{C}_p^\infty(M):\quad f\sim g:\iff f\restriction\equiv g\restriction$$ and the ...
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Lie Group: Lie Algebra Structure

Application For groups of endomorphism one has: $$\mathfrak{gl}(V)\cong \langle GL(V)\rangle,\,\mathfrak{sl}(V)\cong\langle SL(V)\rangle,\,\mathfrak{so}(V)\cong\langle SO(V)\rangle,\,\ldots$$ ...
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2answers
38 views

hyperbolic geometry and affine transformations

A very famous geometer recently commented to me that "hyperbolic geometries are the only geometries invariant under affine transformations". It is unclear to me what this comment even means. Can ...
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1answer
15 views

Diffeomorphism between open sets of half-space

Let $\mathbb{H}^{m}=\left\{(x_{1},...,x_{m})|x_{m}\geq0\right\}$. How can i prove that if $A$ and $B$ are respectively open set of $\mathbb{H}^{m}$ and of $\mathbb{H}^{n}$, with $n\ne m$, then they ...
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10 views

Approximating the Arc Length of a Regular Curve with a Broken Line

Question: Suppose $\alpha:[a,b]\to\mathbb{R}^3$ is a regular curve segment. Prove that, for every $\epsilon>0$, there exists $\delta>0$ such that, for any partition ...
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2answers
33 views

Are all differentiable curves injective?

I'm working through a Differential Geometry text. The author makes a statement I'm having a hard time understanding the validity of. He defines a curve in $\mathbb{R}^3$ as a diffentiable function ...
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What is the covariant basis around a schwarzschild black hole? [migrated]

First of all, I'm not interested in time for this question. So lets consider a 3-manifold whose metric is the spatial part of the Schwarzschild metric, so it has the event horizon and the singularity ...
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2answers
43 views

Can someone explain the basic idea behind the sectional curvature formula?

I found the following equation on Wikipedia here: \begin{equation} K(u,v)={\langle R(u,v)v,u\rangle\over \langle u,u\rangle\langle v,v\rangle-\langle u,v\rangle^2} \end{equation} No explanation I ...
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1answer
33 views

a valuable question about differential geometry, the curve$\beta(s)=\alpha(s)-rn(s)$

Let $\alpha(s)$,s$\in [0,l]$ be a closed convex plane curve positively oriented. The curve $$\beta(s)=\alpha(s)-rn(s)$$,where r is a positive constant and n is the normal vector, is called a parallel ...
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1answer
82 views

Why is Klein bottle non-orientable?

I am doing the homework of differential geometry and encounter this problem: The Klein bottle $K^2$ is defined to be the identification space $$[0, 1] \times [0, 1]/{\sim}, \text{ where the ...
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1answer
46 views

Visualizing Ricci scalar curvature

I am trying to learn more about Ricci scalar curvature. I am trying to get an image in my head of what scalar curvature actually represents about the curvature of a manifold. The most familiar image I ...
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2answers
37 views

What's the difference between a directional derivative and a derivation?

I asked my uncle what a derivation is and and he wrote the following: Most calculus courses discuss directional derivatives and include geometric applications to surfaces of the form ...
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38 views

Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...
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22 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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1answer
22 views

Flat connection of a vector bundle over a 1 dim. manifold

I'd like to show that a connection of a vector bundle $E$ over a 1 dim. manifold $M$ is flat, or equiv. that its curvature is zero. Let $D$ denote the connection, $\sigma$ a section of $E$ and $v,w$ ...
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1answer
18 views

Projection map from $\mathbb R^n-(0)$ to $\mathbb RP^{n-1}$ is smooth

How do I show that the projection map from $\mathbb R^n-(0)$ to $\mathbb RP^{n-1}$ taking $x$ to its equivalence class $[x]$ is smooth?
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Prove the following statement: [on hold]

If $\hat \zeta \epsilon se(3)$, show that $g\hat \zeta g^{-1}\epsilon se(3)$ where $g\epsilon SE(3)$
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Tangent space of a Product of two manifolds

Suppose $M$ and $N$ are two $C^\infty$ manifolds. Take $p\in M$ and $q\in N$. We have the following maps between these: $\iota_1 : M\to M\times N$, $\iota_2:N\to M\times N$, $\pi_1:M\times N\to M$ and ...
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1answer
21 views

References for mean curvature flow in Riemannian manifolds

I am interested in mean curvature flows (MCFs) in Riemannian manifolds. But most textbooks about MCF seem to treat mainly MCFs of hypersurfaces in $\mathbb{R}^{n+1}$. Should I study them before ...
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How to find the points of self intersection of Cayley's Sextic?

I am given that $Y(t)=\cos^3(t)(\cos(3t),\sin(3t))$ and need to find the unique point of self intersection. So I assumed $$\cos^3(t)(\cos(3t),\sin(3t))= \cos^3(u)(\cos(3u),\sin(3u)).$$ I took lengths ...
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25 views

To Prove that The Level Set Of AConstant Rank Map is a Manifold

Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function of constant rank $r$. Let $\mathbf a\in \mathbf R^n$ be such that $f(\mathbf a)=\mathbf 0$. Then $f^{-1}(\mathbf 0)$ is a manifold of ...
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1answer
67 views

Why do we require that a complex manifold has the structure of a real manifold?

I am taking a course in complex manifolds, heavily influenced by Huybrechts' book "Complex Geometry", and in it we define a complex manifold $X$ to be a smooth, real manifold $M$ together with an ...
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Dynamical systems,forward invariant [on hold]

Show that the complement of a forward invariant set is backward invariant, and vice versa. Show that if f is bijective, then an invariant set A satisfies f t (A)= A for all t. Show that this is false, ...
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3answers
44 views

Does parallel transporting require an ambient space?

Can someone summarize why an ambient space isn't needed to measure curvature when parallel transporting tangent vectors or vector fields along a curve on a Riemannian manifold? How do we define the ...
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Determine the equation of the plane . [on hold]

We have given: $$ 1) \ y= x \\ 2) \ y = x^2 $$ Now, I need to determine the equation of the plane for 1) and 2). Help. Thanks in advance.
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What's the name of a solid that results from extruding an area straight along an axis?

If you have any kind of 2D shape and move it up into the third dimension, what do you call it, because it seems like prism is used only if the base is a polygon. It also seems like extrusion is a ...
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20 views

a question about differential geometry, the relation between osculating plane and the points of $\alpha(s)$

question:please prove the limit position of the circle passing through $\alpha(s)$,$\alpha(s+h_{1})$,$\alpha(s+h_{2})$ when $h_{1}$ and $h_{2}$ approaches 0 is a circle in the osculating plane at s, ...
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2answers
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If a curve $\gamma$ through two points $P,Q$ satisfy $\|Q-P\| = \int^{t_1}_{t_0} \| \gamma^{'} \| \, \text {d}t$, then $\gamma$ is a straight line?

In a theorem called "A straight line is the shortest curve through two given points", I prove that for any two points $P,Q \in \mathbb R^2$ and any curve $\gamma : (a,b) \rightarrow \mathbb R^2$ with ...
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0answers
43 views

What is this inner product on differential forms?

I am trying to understand the definition of $d^\ast$ of $d$ where $d$ denotes the exterior derivative as given in these lecture notes. (please see page 3) Here are my thoughts so far: Let us ...
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1answer
39 views

Is it known whether $S^6$ is a Kähler manifold?

I have just started to learn about Kähler manifolds and I now am wondering: Is it known whether $S^6$ is a Kähler manifold? By definition a Kähler manifold has 3 structures: a symplectic, a ...
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2answers
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Integrating the Riemannian volume form

Let $M$ be a compact manifold with $\partial M = \varnothing$ and let $\omega$ be the volume form $\sqrt{\det g_{ij}} dx_1 \wedge \dots \wedge dx_n$. I want to show that $\omega$ is not exact. My ...
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2answers
41 views

Some question about this proof about Riemannian volume form

In these lecture notes lemma 2.3. is given as $\omega_g = \sqrt{\det g_{ij}} \, dx^1 \wedge \cdots \wedge dx^n$ is independent of the choice of coordinate charts. I am trying to understand the ...
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21 views

Sectional curvature in 3-dimensions

I wonder how to compute the sectional curvature of 3-dimensional objects eg. unit ball, $H=\{(x_{0},x_{1},x_{2},x_{3})\in \mathbb{R}^{4}:x_{0}^{2}-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})=1$ and ...
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Does there exists known special cases of a zero Riemann tensor for 3D metrics?

In two dimensions, if one has a flat metric $g_{ab}$, then one can transform $g_{ab}$ to another flat metric $h_{ab}=e^{2\varphi}g_{ab}$, when $\nabla^2 \varphi =0$ and the Riemann tensor remains ...
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one question about differential geometry,show the curvature k($\phi$)

One often gives a plane curve in polar coordinates by $p=p(\phi)$,$a\le\phi \le b$. (1)Show that the arc length is $$\int_{a}^{b}\sqrt{p^2+\dot p^2}$$,where $\dot p$ means the derivative of p with ...
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47 views

Decomposition of a group manifold; is there an associated group decomposition?

The real symplectic group manifold is diffeomorphic to this Cartesian product of manifolds: \begin{equation} \operatorname{Sp}(2n,\mathbb{R}) \simeq \operatorname{U}(n) \times \mathbb{R}^{n(n+1)}. ...
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1answer
44 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
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1answer
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How to smoothly extend a function?

Here is what I am trying to do: Let $X$ be a paracompact smooth manifold. Let $C$ be closed, $U$ open and $C\subset U \subset X$ and $f$ is a smooth map on $U$. I want to show that then there ...
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1answer
67 views

What is the step in this proof “because $\omega$ is closed”?

I am working through this proof of the Poincare lemma here but I don't understand one step. First, there is the following equation $$ {\partial \over \partial x^j} f(x) = \int_0^1 \left (t ...
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1answer
32 views

How to prove that volume forms agree on $U_\alpha \cap U_\beta$?

I am familiarizing myself with Riemannian manifolds. Let $M$ be an orientable smooth $n$-manifold with atlas $(U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$ and let $g$ be a Riemannian metric on $M$. ...
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74 views

Show $f:\mathbb{R}^n \to \mathbb{R}^m$, $n>m$ can't be 1-1

Problem 2-37 on p. 39 of Spivak's Calculus on Manifolds asks Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a continuously differentiable function. Show that $f$ is not 1-1. (Hint: If, for example, ...
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1answer
40 views

When is a curve parametrizable?

Is there a way in general to tell whether a given curve is parametrizable?
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26 views

Tangential derivative vs covariant derivative

My question is basically the same as this, but the answer in that page was not clear to me. Let me restate the question here: let $\Omega\subset\mathbb{R}^3$ be a domain with boundary $\Gamma$, and ...