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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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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88 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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58 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 ...
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23 views

$\beta = b_i y^i$ be a one form and $s_{ij} =\frac{1}{2} (b_{i:j} - b_{j:i} )$

Let $\beta = b_i y^i$ be a one form and $s_{ij} =\frac{1}{2} (b_{i:j} - b_{j:i} )$, where : is the covarient differentiation with respect to Levi- Civita Connection. Then what is the value of ...
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62 views

How do I flow in a circle on a set of vector fields?

Consider a set of complete vector fields on a manifold. They each have an associated one parameter group of diffeomorphisms related to the generated flow. What is a necessary and sufficient ...
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58 views

We can get the geodesic only by the first geodesic equation in hyperbolic plane?

The two geodesic equations are: $$\frac d{ds} (Eu' + Fv') = 1/2(E_u {u'}^2 + 2F_u u'v' + G_u {v'}^2) $$ $$\frac d{ds} (Fu' + Gv') = 1/2(Ev{u'}^2 + 2F_v u'v' + G_v{v'}^2) $$ And in the hyperbolic ...
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42 views

Map projection for the surface $z=xy$ onto $\mathbb{R}^2$

(Hope I used the right tag; please correct if not.) Clarification: In the title, I mean "map projection" in the geographical sense of projecting the surface of a sphere onto a plane, but instead of ...
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44 views

Space generated by a reflection

Suppose I embed a mirror (not necessarily plane) in some space (say a manifold). Is there a theory that tells you how to classify the "space" generated by the reflection (the one you see if you were ...
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27 views

Poincare inequality on balls to general open subset

Let $M$ be an n-dim compact Riemannian manifold with $Ric \geqslant -(n-1)$, it's well known that the following Poincare inequality holds for any function $f\in W^{1,2}(M)$ $$ (\int_B ...
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77 views

The change of parameter of a regular curve is a diffeomorphism, and preserves the length

Let $C$ be a regular curve and let $\alpha:I\subset\mathbb{R}\to C$, $\beta:J\subset\mathbb{R}\to C$ be two parametrizations of $C$ in a neighborhood of $p\in\alpha(I)\cap\beta(I)=W$. Let ...
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35 views

Uniqueness of backward heat equation on closed manifold with given initial data

Suppose we have a closed (compact without boundry) Riemannian manifold $(M,g)$, do we have uniqueness (assuming solutions exist on $[0,T]$ ) for the backward heat equation $$\frac{\partial f ...
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47 views

Showing $\dim T_0 ℝ^n = n$ using a derivation definition for the tangent space.

I’m trying to (re-)prove that $\dim_ℝ \mathrm{Der}_ℝ(C^1(ℝ^n)) = n$, where $$\mathrm{Der}_ℝ(C^1(ℝ^n)) = \{δ\colon C^1(ℝ^n) → ℝ;~\text{$δ$ is a $ℝ$-linear derivation}\},$$ and the ...
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35 views

Proving that $\frac{d}{dt}\int \Phi_t^*\omega=\int_{\Phi_t \circ \partial c} i_{\mathbb{X}}\omega$

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k \rightarrow \mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show ...
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63 views

Order of Riemann tensor indexes and the Ricci Identity

I have seen the Ricci identity written variously as $R_{ijk}{}^l x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R_{ij}{}^l{}_k x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R^l{}_{kij} x^k = ...
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335 views

Partial Differential Equations Course And Differential Geometry Prerequisites

Is the ordinary differential equations course a prerequisite for the partial differential equations course for a person who has passed the integral calculus course? Is it really required to have ...
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29 views

An example of regular singular points

I was reading a book on differential geometry and after the intro to the concepts of regular singular points I came across an example under it: The set $M:=\{(x^2,y^2,z^2,yz,zx,xy)|x,y,z\in ...
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53 views

Variation of geodesic and Jacobi field

I was reading on Jacobi field of a geodesic, and noticed that given a geodesic $\gamma$ it is defined using the term of variation or family of geodesics $\gamma_s$ but never mentioned how to create ...
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60 views

Poincaré Lemma problems and computing contractions in an economical way

Let $x=(A,B,C,D)$ be coordinates on $\mathbb{R}^4$. $\displaystyle \beta = \frac{(AdB-BdA)\wedge(dC \wedge dD)+(dA \wedge dB)\wedge(CdD-DdC)}{(A^2+B^2+C^2+D^2)^2}$ I would like to compute ...
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94 views

Identity of the pushforward of a vector field using a Jacobi bracket.

Let $Z(u,v)$ be the vector field $Z(u,v)=(u^2+u,v^2+v)$, let $\Gamma_t$ denote its flow. I have shown that $[X,Z]=Z-X$. Show that $(\Gamma_t)_*X=e^{-t}X-(e^{-t}-1)Z$. Could someone please show me ...
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22 views

Degree of a map $\phi: R^{n} \rightarrow S^{n}$

I've read a few papers in which they state that the winding number of a mapping $\phi: R^{3} \rightarrow S^{3}$ can be written as the integral $$\int_{\mathcal{R}^3} \epsilon_{ijk}\epsilon^{abc} ...
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Animation/deformation among developable surfaces

While making self-study of differential geometry, animation of surfaces showing isometry between Gauss curvature $K<0$ surfaces Catenoid and Helicoid always fascinated me. I often felt that $ K = ...
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31 views

Curvature 2-form vs. Sectional Curvature

I am familiar with these two notions of curvature on a Riemannian manifold. How are they related to each other?
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37 views

What is the formula for $\frac{\partial}{\partial x_j}(f \circ F)$?

Let $F=(F_i)_{i=1}^n: X \to Y$ be a map between two manifolds. Suppose that $(U, x_1, \ldots, x_n)$ is a local coordinate on $X$ and $(V, y_1, \ldots, y_m)$ is a local coordinate on $Y$. Suppose that ...
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58 views

Second fundamental form

Is it correct to say that .. the second fundamental form of surface theory determines the Euler characteristic and the genus of the surface ? If not how is it determined?
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69 views

Find max distance from $(0,0)$ to line defined on ellipse.

I have got a following problem : $E = \{ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1 \}$ $N$ - line (normal) perpendicular to E at $(x_0,y_0)$ Find max $dist(N,(0,0))$ So I am starting with attempt to ...
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51 views

Why aren't those spaces diffeomorphic?

(Taken from Bredon - Topology and Geometry): Let $X$ be the graph of the real valued function $\theta(x)=|x|$ of a real variable $x$. Define a functional structure on $X$ by taking $f \in F(U) ...
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79 views

Differential Geometry for Computer Science

I am looking for a good book or other resources on Differential Geometry for Computer Sciences or more specifically Differential Geometry used in Computer Graphics, Geometric Modelling and Mesh ...
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22 views

For two unit speed curves $\gamma_{1,2}$ on $[0,l]$ where $0\leq k_2\leq k_1<\pi/l$, show $d(\gamma_1(0),\gamma_1(l))\leq d(\gamma_2(0),\gamma_2(l))$.

For two unit speed curves $\gamma_{1,2}$ on $[0,l]$ where $0\leq k_2\leq k_1<\pi/l$, show $d(\gamma_1(0),\gamma_1(l))\leq d(\gamma_2(0),\gamma_2(l))$. It is important to note here that ...
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35 views

Definition of a connection on a principal bundle

I am trying to understand the definition of a connection as given in, for example, Taubes' book Differential Geometry. Let $\pi: P \to M$ be a principal bundle with a $G$ action. He states that a ...
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61 views

Angle of intersection between a plane and sphere.

Let $X(\theta,\phi)=(\sin \theta \cos \phi, \sin\theta\sin \phi, \cos\theta)$ be parametrization of the sphere $S^2$. Let $P$ be the plane $x=z \cot\alpha$, $0<\alpha<\pi$ and $\beta$ be the ...
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36 views

Are there some reference books or handbooks on homology and homotopy groups of every manifold which has been calculated?

Are there some reference books or handbooks on homology and homotopy groups of every manifold which has been calculated ? Or Are there some reference books especially on differential geometry and ...
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45 views

Total variation of real valued functions on a manifold

We can define the total variation of a function $f:\mathbb{R}^n\to \mathbb{R}$ as in Evans and Garriepy's "Measure Theory and Fine Properties of Functions" or in this Wikipedia article ...
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66 views

three vector fields with zero Lie bracket

Suppose I have three unit vector fields $u, v, w$ on a two-dimensional surface in $\mathbb{R}^3$. Are there simple compatibility conditions relating $v$ and $w$ such that there exists scalar fields ...
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26 views

Topology of operator bundle?

I am trying to understand the family version of the Atiyah-Singer Index Theorem as described in the book "Spin Geometry" by Lawson/Michelsohn. In Part III.§8, they define the operator bundle $$ ...
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39 views

How to compute the classic differential of a section

Given a section $s:M\to E$ of a vector bundle, I'm trying to compute its differential, $Ts:TM\to TE$ in a local trivialization. To do this, I can write $$s=f^ie_i$$ For some local frame $(e_i)$ of ...
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Embedding counterexample

Lee writes on page 156 of Introduction to Smooth Manifolds: A smooth embedding is a map that is both a topological embedding and an immersion, not just a topological embedding that happens to be ...
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Calculating Integral Submanifolds

I have the vector fields $v_{1} = x \partial_y - y \partial_x + z \partial_w - w \partial_z$ and $v_{2} = z \partial_x - x \partial_z + w \partial_y - y \partial_w$ on $S^{3} \subset \mathbb{R}^4$. I ...
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26 views

Continuity of length under uniform convergence

Let $(\phi_n):[0,1]\to \mathbb{R}^d$ be a sequence of injective curves, parameterized by constant speed, uniformly converging to a limit curve $\phi:[0,1]\to \mathbb{R}^d$. Assume: $\sup_n ...
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51 views

Christoffel Symbols on a Surface

In Do Carmo's Differential Geometry of Curves and Surfaces he does the following: Let $\vec r$ be a parametrization of a surface $S\subset\mathbb{R}^3$ so that $\vec r_u,\vec r_v$ forms a basis for ...
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36 views

Inner product between basis vectors

Given a vector field $A=A_x \hat x + A_y \hat y$. If we want to represent this vector field with respect to the polar coordinate vector fields $\hat r$ and $\hat \phi$, we'd just perform the dot ...
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Does this lemma have a name or where can I find a proof?

Does the lemma at the bottom of this page have a name? Or could someone give me an idea of where I can find a proof? In case you can't access the link: Lemma $\ \ $ If $g$ is of class ...
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85 views

Characteristic class integral: on what manifold does $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$ hold?

Characteristic class integral: when does the equality hold $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$, on what manifolds? Here $c_1$ is the first Chern class. Here $w_2$ is the 2nd ...
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62 views

Lie Bracket of vector fields on Lie group

Let $H$ be a Lie group and $\mathfrak{h}$ its Lie algebra. Given a smooth function $v: H \to \mathfrak{h}$, define the vector field $\bar{v} : H \to TH$, $h \mapsto d(R_{h})_{e} v(h)$, where $R_{h} : ...
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39 views

Torsion of an asymptotic curve with nonzero curvature

I wish to solve the following problem using the matrix of the shape operator $S_{P}$. Suppose $K= \text{det} S_{P} <0$, and $C$ is an asymptotic curve with curvature $\kappa (P)$ nonzero. I want to ...
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94 views

double Hodge star operator

$*$ is the Hodge star operator acting on differential $k$-forms on $\mathbb{R}^{n}$ as below: $$*:Λ^k(\mathbb{R}^{n}) \to Λ^{n-k}(\mathbb{R}^{n})$$ $$\alpha \wedge (\star \beta) = \langle ...
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34 views

Make differential 1-form invariant for lift to universal cover

Suppose $ F : M \to M$ is diffeomorphism of smooth manifold $M$, and suppose $F^* \nu = \nu$ for differential 2-form $\nu$. Let $p: \tilde{M} \to M$ denote universal cover of $M$, and suppose $p^*\nu$ ...
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34 views

Reference request: foliations

I am looking for a gentle introduction to foliations for smooth manifolds, but I have a hard time finding a textbook explaining this notion. Wikipedia's links are also to articles. Is there any ...
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36 views

homotopy via isotopy

Is it possible to realize a homotopy between two vector bundles (which are sub-bundles of the tangent bundle) over the same base $B$ as induced by an ambient isotpy? In other words, assume $X_t $ is a ...
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17 views

Is $z^2 = x^2 \cos y + 1$ an orientable surface?

So I can find one parametrization $\phi (u, v) = (u, v, \sqrt{u^2 \cos v + 1})$ which only does have of it. So now I need to find another parametrization which overlaps non trivally with $\phi.$ I ...