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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$SO(3)$ has a subgroup $U(1) \times U(1)$?

I am wondering - and asking you - whether there is a subgroup $U(1) \times U(1)$ of the Lie group $SO(3)$. Equivalently, I can reformulate it from a geometrical point of view: does there exist a torus ...
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35 views

Problems in understanding the notion of “active” diffeomorphisms on a manifold

I have been studying differential geometry recently in the hope of gaining a deeper understanding of General Relativity (GR). I have run into an issue when trying to understand the notion of what is ...
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90 views

How to evaluate solid angle subtended by an ellipse at any arbitrary point on the vertical axis passing through the center

`Question: How to evaluate the exact value of solid angle subtended by an ellipse (or elliptical plane) at any arbitrary point lying on the vertical axis passing through the center. Standard equation ...
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49 views

the gradient in complex coordinates

Let $M$ be an $n$-dimensional complex manifold, or equivalently a $2n$ real manifold. Let $g$ be a Riemannian metric. Let $f\in C^\infty(M)$. What is $\nabla f$? If $x_1,\dots,x_n,y_1,\dots,y_n$ are ...
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72 views

Relative version of Whitney embedding's theorem (reference needed)

One form of Whitney embedding theorem says that if $M,N$ are smooth compact manifolds and the dimension of $N$ is more than twice the dimension of $M$, then the space of embeddings $M\to N$ is open ...
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34 views

Uniform convergence of the harmonic form heat flow

[${\bf NOTATIONS}$] Let $M$ be a closed Riemannian manifold of $m$ dimensional, $p\in\{1,\cdots,m\}$. $A^p:=\{\text{smooth p-forms on }M\}$. $\delta:A^{p+1}\to A^p$ denotes the formally adjoint ...
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61 views

Integration on $ \mathbb{P}^n ( \mathbb{R} ) $.

Could you tell me please, when and how we calculate the integral of a function on $ \mathbb{P}^n ( \mathbb{R} ) $ ? Do you have some references about that ? Thank you in advance.
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57 views

confusion about basic Kahler geometry

I am really struggling to understand the basics of Kahler geometry and hope someone can give me some guidance. Suppose we have a complex manifold with some complex structure $J$ and let $g$ be a ...
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58 views

Coding for a calculation in differential geometry using Maple

I am beginner in maple. And my field is Differential geometry. I've learnt lie brackets using maple help. But I am testing this calculation through maple. I have these vector fields $e1=z^2\ast ...
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46 views

Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
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52 views

Direction of outward normal differential $\hat n\ ds$

In vector calculus, why is the outward normal differential $\hat n\ ds$ around a closed curve $dy\ \hat i - dx\ \hat j$? Why isn't it $-dy\ \hat i + dx\ \hat j$ or something else?
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28 views

determinant of metric on complexified vector space

Let $M$ be a complex manifold of dimension $n$ with Hermitian metric $g$. Extend $g$ to $TM^{\mathbb{C}}$ linearly. I believe that at each $p$ we can say a basis for $T_pM$ as a real vector space is ...
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66 views

Closed 1-forms on Simply Connected Manifold

Is it true that closed 1-forms on a simply connected differentiable manifold are exact. If so, could you explain why? Thanks very much
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18 views

When is the composition of multiple flows the identity?

Suppose I have some number of vector fields $a,b,\ldots,k$, and I denote flow along $a$ for time $t$ by $a_t$, etc. When is it the case that for all points $x$, $a_tb_tc_t\ldots k_tx = x$? In the ...
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38 views

Linearisation in direction of formal adjoint

Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. The Ricci curvature can be viewed as a differential operator ...
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28 views

Lie Groups map question

This is a question about Exercise 5.59 in Jeff Lee's Manifolds and Differential Geometry. He writes For a fixed $A\in GL(V)$, the map $L_A:GL(V)\rightarrow GL(V)$ given by $A\mapsto A\circ B$ has ...
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30 views

Minimising surface with given curve as a boundary

I have a problem connected to finding a minimal surface with a given boundary. I know that it is the surface with zero mean curvature but as I have to obain differential equations for such surface I ...
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64 views

Flat vector bundles and constant transition functions

Let $E\to M$ be a vector bundle endowed with a flat connection. Then, does $E$ admit a bundle atlas with constant transition functions? For a vector bundle with constant transition functions, are ...
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21 views

Plane fields transversal to 1-dimensional bundles and Ehresmann connection

This is a question related to something I saw in the book Confoliations by Thurston and Eliashberg. Consider three G-bundles $M \rightarrow F$ and two dimensional plane fields $\xi$: A. $M= F\times ...
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71 views

Integration over a second order tensor

I would like to compute the mean value of a second order tensor $\mathbf{T}$ expressed in planar cylindrical coordinates. The mean value for any second order tensor is (reference [1] page 101) ...
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27 views

(Proof Check) Orientability of zero set of smooth function

Could you check my proof that zero set of a smooth function on Euclidean space is an orientable manifold? Suppose $g:R^{n-k}\times R^k\to R^k$ is smooth such that $\text{rank}g'(x)=k$ whenver ...
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44 views

Schwarzschild half-plane and its geodesics

For some fixed $r_0>0,$ put the semi-Riemannian metric $$ ds^2=\frac{r_0-r}{r}dt^2+\frac{r}{r-r_0}dr^2 $$ on $\{(t,r)\in\mathbb{R}^2:r>r_0\}.$ I would like to show that the $r$-lines are always ...
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55 views

Gradient curve of a harmonic function

I am reading the paper "Energy of Harmonic function and Gromov proof of Stalling theorem" https://www.math.ucdavis.edu/~kapovich/EPR/energy.pdf I have no clue about the lemma 8.4(i). What is gradient ...
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31 views

Proof that Inversion changes the direction of the normal vector.

The question I am working with is to show that inversion, defined by $F(q) = \frac {q}{|q|^2}, q \in \mathbb R^n$ preserves the unit $n$-sphere, but changes the direction of the normal vectors. That ...
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53 views

Computing the tangent map using partial derivatives

Let $G$ be a Lie group and let $\mu: G\times G \rightarrow G$ be a smooth map. I want to compute the tangent map $T_{(e,e)}\mu: T_{e}G\times T_{e}G\rightarrow T_{e}G$ of $\mu$. In a proof in my notes ...
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92 views

Prove that $f$ is a diffeomorphism

Let $$E = \left\{ (x,y,z) \in \mathbb{R}^3: \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \right\}$$ for some $a,b,c > 0$. Prove that the function $$f:E \to S^2, \quad (x,y,z) \mapsto ...
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100 views

Existence of a holomorphic connection implies existence of a flat connection

Let $E$ be a holomorphic vector bundle on a complex manifold $X$, and let $D: E \to E \otimes \Omega_X$ be a holomorphic connection on $E$ i.e. $$ D(fs) = s\otimes \partial(f)+ fD(s), $$ for any local ...
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90 views

Question about Milnor's proof of Sard's Theorem

We've just covered Sard's theorem and have just started to look at transversality in my differential geometry class and I'm trying to understand a proof of Sard's theorem (based on Milnor's proof): If ...
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35 views

Open cover of manifold with boundary

If $\mathcal{O}=(U_{\alpha})_{\alpha\in A}$ is an open cover for a smooth manifold $M$, then each $U_{\alpha}$ is a smooth manifold. I want to extend this fact to manifolds without boundary. So my ...
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43 views

Interpreting the Contravariant derivative being zero

I'm doing some calculations for Riemannian manifolds and I'm having trouble understanding one of the results. Under certain conditions, I've shown that for an antisymmetric rank two tensor $F_{ij}$, ...
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64 views

Lie Derivative of a section on a vector bundle

I'm still trying to figure out how to do the Lie derivative of a Jacobian. (c.f. earlier unanswered post). If I know how how to do Lie derivatives on section of vector bundles, that would be ...
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72 views

Calculate Parallel transport

Suppose I want to parallel transport a vector $v$ living in the tangent space of my surface at a point $p$ along a closed geodesic polygonal path. On each regular component I keep the angle between ...
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58 views

Calculating Normals across a sphere with a wave-like vertex shader

This is a bit of a CS question, but more than not it's a 3D math problem. I've been trying to get the correct normals for a sphere I'm messing with using a vertex shader. The algorithm can be boiled ...
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61 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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31 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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82 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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136 views

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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95 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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30 views

Parametrization of surfaces gauss

What is pde relating $f$ and $g$ if the Gauss curvature is $+1$ and $-1$ respectively: $$ \begin{align*} (x,y,z) &= ( u \cos(v), u \sin(v), f(u,v) ) \\ (x,y,z) &= ( u \cos(v), u \sin(v), ...
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29 views

Zero Gauss curvature surfaces spanning a loop

EDIT 1: Spanning across a given arbitrary closed boundary/loop a surface can be defined with zero mean curvature H in 3-space as minimal surfaces. Likewise can a surface be defined with zero Gauss ...
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53 views

Basic Differential geometry: Shortest path between two points in R^3 is straight.

Given two points P and Q in $\mathbb{R^3}$, we want to show that the shortest distance between them is through a straight line. let $c(a) = P$ and $c(b) = Q$ and $c(t)\neq P$ for $t>a$(One ...
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44 views

De Rham Cohomology of the complement of an ellipse

I'll call $A$ the complement of an ellipse in $\mathbb{R}^2$. I know that $H^0(A)=\mathbb{R}^2$ and $H^k(A)=0$ if $k \ge 3$. What about $k=1,2$? I know that $A$ has two connected components, so ...
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56 views

Identity involving Riemann tensor

I'm reading about the Ricci tensor, and I've found the following statement that is given without proof: For a point $p$ on a Riemannian manifold, and coordinate vector fields $X_{\alpha}$, ...
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79 views

Definition of the second fundamental form in $\Bbb L^3$, following Kühnel.

I'm having trouble understanding a few comments is Kühnel's "Differential Geometry: curves - surfaces - manifolds", in page $118$, about the definition of the second-fundamental form in ...
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25 views

Why do entries of Maurer-Cartan 1-form span space of left-invariant 1-forms

Suppose $G$ is a $k$-dimensional Lie group of $n\times n$ matrices of the form $[g_{ij}(x_1,...,x_k)]$ where $g_{ij}$ are smooth functions. As a follow-up to the question I posted recently, I now ...
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70 views

Harmonic maps and angle preservation

I have the following question: What are the angle preservation properties of harmonic maps? Conformal maps preserve angles exactly, but they distort lengths. In this sense a conformal map is the ...
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75 views

Explicit Calculation of the Euler class for the 2-Sphere using transition functions

I have been trying to learn about characteristic classes for months now, and every time I try a simple example something goes wrong. Any insights would be greatly appreciated. I am trying to follow ...
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83 views

Integral Curve of the vector field

If we have a 2-sphere with coordinates $x=r \cos \theta \cos \phi$, $y= r \sin \theta \sin \phi$ and $z=r \sin \theta$ and the vector field $X= (-r\sin \theta \cos\phi, r \cos \theta \sin \phi, r \cos ...
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57 views

Lie Derivative of the Jacobian?

$X,Y$ smooth manifolds. Given a map $f:X\longrightarrow Y$. Let's say for simplicity that $f$ is a diffeomorphism and defined everywhere. Let $v,w$ be smooth vectorfields on $X$ and $g\in\mathcal ...
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39 views

Asymptotic lines on $ x^2 y^2 + y^2 z^2 + z^2 x^2 = 1 $

How are asymptotic lines defined/computed on this implicit surface? EDIT1: In Monge form $ z = \frac { 1- z^2 y^2}{x^2+y^2}, $ I find later it can be done because it can at all be cast into such ...