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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A proof of exactness of closed 1-forms on the two-sphere

Remark. I'm aware that here the same problem is solved. I'm in trouble with the proof I'm going to quote. Consider the following Claim. Every closed 1-form $\beta$ on $S^2$ is exact. This is an ...
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94 views

Boundary on a manifold

I was wondering how I can see if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I need to ...
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126 views

differential geometry : basic query about tensor notation and tensor products

I have a few very basic queries. I've been studying differential geometry as part of a course on General Relativity, so I don't have a very well grounded understanding of the mathematical formalism; ...
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80 views

Lie Groups: Differential Operations

Given a Lie group. Multiplication and inversion act infinitesimally at the identity by: $$\mathrm{d}\mu:\mathrm{T}_{(e,e)}(G\times G)\to\mathrm{T}_eG:(u,v)\mapsto u+v$$ $$\mathrm{d}\iota:\mathrm{T}...
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An algorithm to find isometry between surfaces in $\mathbb{R}^3$?

Given two surfaces in $R^3$, i would like to find isometry between these two. Usually, in class, we did some examples, like bending the plane into a cylinder, or cone, and they were not hard, quite ...
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193 views

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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53 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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80 views

Left-Invariant Vector Fields: Smoothness

Given a Lie group $G$. For a left-invariant vector field it holds: $$\mathrm{d}l_gV=V\circ l_g:\quad V_g=\mathrm{d}l_gV_e$$ Conversely rough vector fields are smooth: $$V_g:=\mathrm{d}l_gv:\quad V\...
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745 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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57 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)= \frac{1}{\sqrt{2+e^{12t}...
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195 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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57 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 two-...
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103 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
64 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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75 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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2answers
180 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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76 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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471 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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227 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
227 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 $G(x,y,z)=0$,...
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85 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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101 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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42 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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42 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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51 views

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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60 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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53 views

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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1answer
157 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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3answers
67 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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25 views

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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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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246 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
51 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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66 views

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
85 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 proof....
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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 zero....
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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 ...
2
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78 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
126 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 $L = -DN \circ Df^{-1}.$ ...
2
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1answer
91 views

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
75 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
48 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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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
58 views

When is a curve parametrizable?

Is there a way in general to tell whether a given curve is parametrizable?
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192 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 ...
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2answers
84 views

Notation of coordinate representation in Lee

In Lee's Introduction to Smooth Manifolds he writes $$ \omega = \omega_i dx^i$$ where $\omega$ is a differential form. See for example page 293. What does $\omega_i dx^i$ stand for? According ...
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61 views

Is my understanding of the argument correct?

I worked through a proof of: $$ f(z) = {1\over 2 \pi i}\int_{\partial D} {f(w) \over w -z} dw$$ where $D\subset \mathbb C$ is an open disk and $f$ is holomorphic on $D$ and continuous on $\overline{...
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68 views

Name of isometric invariant in Gauss-Bonnet

Does the tangential rotation term $ \int k_g ds $ of Gauss-Bonnet theorem ( for continuous or discontinuous lines on a surface) have a name or symbol in differential geometry ? The second term $ \...
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1answer
60 views

Can I argue like this to prove that the determinant is positive?

Let $X$ be a smooth $n$-manifold with an oriented atlas $\mathcal U = (U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$. Let $g$ be a Riemannian metric on $X$. Let $g_{ij} = g\left ( {\partial \over \...
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3answers
317 views

Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} \text{div}X=-g^{ij}g(\nabla_iX,\partial_j)...