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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Geodesic Lines on Covering Maps

So I'm not sure how deck transformations work into this problem. I've established the following so far. Let $\pi:\tilde{M}\rightarrow M$ be the universal covering map. We may suppose that $M$ is ...
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42 views

Constructing triangulations algorithmically

I am developing a Python package for computations in algebraic topology (namely: cohomology and Massey products on manifolds). Basically all the stuff I'm doing requires an explicit triangulation of ...
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39 views

How One Can Find the Envelope from Parametric Equations?

How to generally find singular solution or envelope of a two parameter family of curves $ x(u,v),y(u,v) $ in the plane? The parametric equations $$x(u,v) = \cos^2 (u) \cos (v) + \cos (u) \sin (u) ...
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Flow into a tetrahedron

This problem is from Harold Edwards' Advanced Calculus: A Differental Forms Approach. It is exercise $4c$ in section $1.3$. For a unit flow in the $z$-direction find the total flow into the ...
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36 views

definition of tensors and its connection to examples of tensors

Tensors are often introduced using tensor products or multilinear maps. I think I understand how they hang together. The examples given (see https://en.wikipedia.org/wiki/Tensor) are then a bit ...
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Geodesics in a hyperbolic plane like space

For $|\rho| < 1$ and $\sigma >0$ consider the Riemannian metric \begin{equation} g:= \begin{pmatrix} \frac{1}{\left( 1-\rho^2 \right)y^2} & \frac{-\rho}{\sigma\left( 1-\rho^2 \right)y^2} \\ ...
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145 views

Prove the tangent space at a point $x$ of the $n$-sphere is the space $\{v \in \mathbb{R}^{n+1} : v\cdot x=0\}$

I can see why this is true but I'm not sure how to prove it, any help would be appreciated. Prove that the tangent space $TS^{n}_{x}$ at a point $x$ on the $n$-sphere $S^{n}:=\{x \in ...
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Charts for $\mathbb S^1$ exercise.

So here is the following exercise: For $\theta \in R$ we define $U_{\theta}:=\mathbb{S}^1 \setminus\{\cos(\theta), \sin(\theta)\}\subset \mathbb S^1$ and $$\varphi_{\theta}:U_{\theta} \rightarrow ...
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52 views

Winding Number of a Circle

I'm having a little trouble calculating the winding number of a circle about a point using parametric equations. The definition of a circle of radius $r$ and center coordinates $x_0$ and $y_0$ is the ...
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94 views

Show that the centre of the circle $\dots$ approaches the point $\epsilon(s_0) = \dots$ - Elem Diff Geo Pressley

Help on getting started with this exercise Another approach to the curvature of a unit-speed plane curve $\gamma$ at a point $\gamma(s_0)$ is to look for the `best approximating circle' at this ...
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30 views

Involute of a Curve

A string of length ℓ is attached to the point γ(0) of a unit-speed plane curve γ(s). Show that when the string is wound onto the curve while being kept taught, its endpoint traces out the curve ι(s) = ...
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32 views

condition of being regular surface

Let $S_a$ be the surface given by $x^4-x^2+y^2+z^2=a$ Let $F=x^4-x^2+y^2+z^2$ $(F_x,F_y,F_z)=0$ when $x=0$ or $ \pm \frac{1}{\sqrt{2}}$ So I guess $S_a$ is regular when $a \ne 0$ and $-\frac{1}{4}$ ...
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50 views

Periodic solutions of an ODE on a circle.

I am trying to mark some homework solutions. The question is: suppose $\dot{x}=f(x)$, with $x \in \mathbb{S}^{1}$, and $f(x) \neq 0$. Assume existence and uniqueness theorem is satisfied. Is it ...
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22 views

Prove that the map $F:\mathbb{R}\to S^1$, $F(t)=(\cos t,\sin t)$ is $C^\infty$.

Why isn't smoothness of $(\cos t,\sin t)$ as a map from $\mathbb{R}$ to $\mathbb{R}^2$ enough? IS it because when we have $S^1$, the codomain is changed? But that shouldn't affect smoothness, ...
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38 views

Does diffeomorphism act transitively on a $\mathbb B^n$?

I want to find a diffeomorphism F : $\mathbb B^n$ → $\mathbb B^n$ such that F(p) = q for all p,q$\in$ $\mathbb B^n$. Here $\mathbb B^n$ is the just unit open ball in $\mathbb R^n$. My idea is let ...
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28 views

Compute of metric and curvature under transformation of coordinates.

Let $\widehat{g}_{ij}(x,t)$ be a solution of $$\frac{\partial g_{ij}}{\partial t}=-2R_{ij},$$ and $\varphi_t:M\rightarrow M$ is a family of diffeomorphisms of $M$. Let ...
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A question about a function used to define the length of a curve in Riemannian geometry

Let be $(M,g)$ a connected Riemannian manifold. If $ \phi : [a,b] \rightarrow M$ is $C^\infty$ we define the arc-length of the curve $\phi$ as the quantity: $$J(\phi )= \int_a^b f(\phi ...
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16 views

Calculate Ricci Tensor (Axial Symmetry)

I have a problem calculating a Ricci tensor. My metric (Lorentz signature) is $\mathbf{g}=Xd\phi^2 + g_{ab}dx^adx^b$ where $X,g_{ab}$ don't depend on $\phi$, and $g_{ab}$ is a $2+1$ metric. I define ...
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18 views

Regularity of map

Let $M$ be a smooth manifold and $f: M \rightarrow \mathbb{R}^n$ be an immersion of regularity $C^{k, \alpha}$ for $k \geq 2$. Suppose that $U$ is a neighborhood of $M$ and $\chi$ is a vector field in ...
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19 views

Why are these conditions necessary?

I'm asked the following question: Let $\alpha:I\to \mathbb{R}^3$ be a regular, injective curve with everywhere non-zero curvature. Assume that $M$, the image of $$r:I\times\mathbb{R}_+, ...
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44 views

Quadratic differentials and flat metrics in genus 1

Let's write $\mathcal{M}_1$ for the moduli space of Riemann surfaces of genus 1, i.e. the set of isomorphism classes of Riemann surfaces of genus 1. I know it has the following equivalent definition. ...
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29 views

Maximal ellipse

Let $C$ be a closed, strictly convex, smooth, origin-symmetric curve such that its reflection with respect to a line $l$ passing through the origin is itself again. Is it true that the maximal ...
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42 views

A Proof of the Hausdorffness of the Grassmannian Using the Basics

$\DeclareMathOperator{\Span}{span} \newcommand{\R}{\mathbf R} \newcommand{\mc}{\mathcal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\grassman}{GR} \newcommand{\set}[1]{\{#1\}} ...
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prove iff conditions on $(M_1,g_1)$, $(M_2,g_2)$ so that $\big(M_1\times M_2,g_1+g_2\big)$ satisfies $\text{Ric}^0=0$

Problem Show that a product metric $\big(M_1\times M_2,g_1+g_2\big)$ satisfies $\text{Ric}^0=0$, (where $\text{Ric}^0$ is the traceless Ricci tensor) if and only if both original metrics are ...
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20 views

Concept help: Cylinder is a regular surface

Show that the cylinder $\{(x,y,z)\in \Bbb{R}^3:x^2+y^2=1\}$ is a regular surface, and find parameterizations whose coordinate neighborhoods cover it. How can I find the parameterization ...
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51 views

The Hessian at a critial point $p$

In my studies I've come across the hessian in the context of Riemannian geometry. I use the following definition of the hessian $$ H^f(X,Y)=XYf-(D_xY)f=\langle D_X(\operatorname{grad} f),T\rangle. $$ ...
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38 views

(Very) Twisted bundles

Suppose that $M$ is a smooth manifold. Over each coordinate chart $(U,\varphi)$ $M$ looks like euclidean space. In particular the tangent bundle $TM$ is trivial over $M$. So you the covering of $M$ by ...
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49 views

Difference between Euclidean and Hyperbolic lengths.

What is the difference between Euclidean and Hyperbolic lengths? For instance if i were to measure a curve on with the euclidean distance and alternatively the Hyperbolic distance, What would be the ...
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40 views

Domain of integration

I am reading through Lee's Introduction to smooth manifolds and he defines a domain of integration to be a bounded subset of $\mathbb{R}^n$ whose boundary has measure zero. He then goes on to say ...
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27 views

What is the metric of the Riemann surface resulting from quotiening the upper half plane by a Fuchsian group?

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. What is the metric we get on $\mathbb{H} / \Gamma$?
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48 views

Are two isometric Riemannian manifolds necessarily related by Euclidean motions?

Let $(M,g), (\overline{M},\overline{g})$ be smooth Riemannian manifolds that are isometric, i.e. there is a smooth function $f: M \rightarrow \overline{M}$ such that d$f(g)=\overline{g}$. By Nash's ...
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35 views

A compute of Riemannian Geometry.

$M$ is a Riemannian manifold ,and $g_{ij}$ is Remannian metric. Let $x=(x^1...x^d)$ $(i.e. x:U_x\rightarrow R^d)$ be a local coordinates ,and $v,w\in T_pM$ with coordinate representations ...
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44 views

Covariant Derivative Along Arbitrary Vector

The inspiration for this question is section 1.3 of "A Course in Minimal Surfaces," by Colding and Minicozzi. This section has to do with deriving the first variation formula. We are dealing with an ...
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24 views

The Need for Tangent to a curve at a point and its definition

I was understanding derivative function when I thought that why "concept of tangent", was invented.If it was so because of influence of Physics - instantaneous velocity and other stuff then why ...
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51 views

Weird notation in do Carmo's Riemannian Geometry

In do Carmo's Riemannian geometry, in ch. 0, sec. 4, ex. 4.2, he discusses the generalization of a 2-surface in $\mathbb{R}^3$ to a $k$-surface in $\mathbb{R}^n$, $k\leq n$ by defining a subset ...
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Exactly why is a(t) = (t^3, t^2) is not an immersion? What do the vectors in the image of Da(t) look like?

Let $\alpha:\mathbb{R} \rightarrow \mathbb{R}^2$ be a differential mapping defined by $\alpha(t) = (t^3, t^2)$. I'm pretty sure I understand why $\alpha(t)$ is not an immersion, but I really want to ...
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39 views

How to prove that a manifold with Self-Dual Riemann tensor is Ricci-flat

By self-dual I mean that \begin{equation} *\mathcal{R}_{ab}=\mathcal{R}_{ab}\,, \end{equation} where $\mathcal{R}$ is the curvature 2-form, related to the Riemann tensor $R$ by $\mathcal{R}_{ab}= ...
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38 views

Showing function is a diffeomorphism

Let $\mathbb{S_1}$ and $\mathbb{S_2}$ be two regular surfaces with a mapping $f:\mathbb{S_1}\to \mathbb{S_2}$ which I have shown to be bijective from $\mathbb{S_1}$ to $\mathbb{S_2}$. From the ...
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29 views

Poincaré-Birkhoff theorem in sympl. geometry

On p. 274 of McDuff and Salamon's Introduction to symplectic topology a corollary to the Poincaré Birkhoff theorem is presented. So we are given an area preserving map on an annulus ...
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44 views

Does a horizontal bundle on a fibre bundle induce an affine linear connexion on its tangent bundle?

Let $\pi:E\to B$ be a (locally trivial, and smooth) fibre bundle whose fibres are diffeomorphic to $F$, and $\mathcal{H}\subset TE$ be a subbundle satisfying $TE=\mathcal{V}\oplus\mathcal{H}$, where ...
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On closed, symplectically embedded surfaces of ambient compact symplectic manifolds, how does one avoid pulling back a 2-form to an exact 2-form?

Obviously, if one pulls back an exact (with respect to the de Rham d) differential form by any map, then one obtains an exact form on the submanifold. But if one starts out with a form that isn't ...
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Gauss Curvature Equation for hypersurface in Semi-Riemannian manifold

We have the Gauss curvature equation: $$\langle R(V,W)X,Y\rangle = \langle R'(V,W)X,Y\rangle - \langle II(V,X),II(W,Y)\rangle + \langle II(V,Y),II(W,X)\rangle$$ Here $M$ is an immersion in $N$. ...
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Is the induced volume on submanifolds unique?

consider a $2n$-dimensional manifold $\mathcal{M}$ With a volume element $\omega$. Now consider a $(2n-2)$-dimensional submanifold $\mathcal{N}$. How one can define a volume on $\mathcal{N}$ based on ...
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70 views

Proving the Leibniz Rule for Lie Derivatives of tensor fields.

I am learning some Differential Geometry on my own in preparation for a course I'm starting in October, and one of the exercises in the notes I'm using is to check that the Lie Derivative satisfies ...
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36 views

Inverse Function Theorem: is this true?

The inverse function theorem is usually stated as follows: Let $f:\mathbb R^n \to \mathbb R^n$ be a smooth map and let $x_0$ be a point such that $\det J_f (x_0) \neq 0$. Then there exists an open ...
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51 views

Identity regarding the components of a dual basis

This problem is from Robert Wald's "General Relativity." The problem is 4(b) from chapter 2. Let $Y_1\cdots Y_n$ be smooth vector fields on an $n$-dimensional manifold $M$ such that at each $p\in M$ ...
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58 views

Prove the differential of $f$ at $p$ is a well defined linear map

Let $W$ be a subset of $\mathbb{R^n}$ be open and $f:W \rightarrow \mathbb{R^m}$ be smooth. The differential of $f$ at $p \in W$is a linear map: $$df_p: \mathbb{R^n} \rightarrow \mathbb{R^m}$$ ...
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65 views

The second fundamental form of the sphere

I am trying to understand how one computes the second fundamental form of the sphere. Looking at the example on page 10. http://www.math.miami.edu/~galloway/dgnotes/chpt5.pdf Here I understood ...
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48 views

Orientability and volume form

I was wondering if there is an easy argument that manifolds that are orientable always have a volume form and vice versa? So I am not looking for a full proof of this, but rather a good argument how ...
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78 views

gluing manifolds along boundaries

I have a problem with the following task. Suppose $M_1, M_2$ are smooth manifolds with boundary. Let $f_1, f_2$ be diffeomorphisms from $B_1$ to $B_2$ ($B_i$ - the boundary of $M_i$), and suppose ...