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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Problem of Arnold's book and covering spaces

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
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36 views

Preimage of a small normal deformation under an embedding again a normal defomation?

Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a ...
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60 views

Kinds of isometries preserving the curvature tensor

We talked about isometries /local isometries and linear isometries in the lecture, but unfortunately we did not say when the isometry preserves the curvature tensor or sectional curvature. First I am ...
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38 views

Vector field from group action

Let $\Phi: G \times \mathbb{R}^4 \rightarrow \mathbb{R}^4$ be a group action where $G = \mathbb{R}/(2 \pi \mathbb{Z}).$ Then $$\Phi(\theta,(x_1,x_2,p_1,p_2)) = ( R(\theta) (x_1,x_2)^T, R(\theta) (p_1,...
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83 views

A variational approach to symplectomorphisms

Let $(\Sigma,\omega)$ be a compact symplectic surface, and let $\mathcal{D}$ be the group of diffeomorphisms of $\Sigma$. Is there some known variational approach to determine if an element $f\in\...
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62 views

“Commutation” of parallel transport with covariant derivative and Riemann curvature tensor

I'd appreciate it a lot of you could please give me a detailed answer to this question. Alternately, you could just cite a reference too. Let $(M,g)$ be a Riemannian manifold, $c$ a curve on it. You ...
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52 views

Change of variable and diffeomorphic surfaces?

Suppose two curves $\gamma$ and $\gamma'$ are diffeomorphic. Is the arc-length measure $ds_\gamma$ absolutely continuous to $ds_\gamma'$ with a positive derivative? ($ds_\gamma=\phi\, ds_\gamma'$ for ...
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63 views

Second contracted Bianchi identity

Considering the Riemann tensor and the onece contracted second Bianchi identity $g^{ls}\nabla_sR_{ijkl}=-\nabla_iR_{jk}+\nabla_jR_{ik}$ why should it hold true that $g^{ls}\nabla_sR_{ijkl}=0$? In ...
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84 views

Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that $...
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29 views

Bounding distance between geodesics in manifolds with nonpositive curvature

I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I would like to see proven (and clarified). Let $M$ be a compact, connected ...
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33 views

For which values of $c$ are the following sets become smooth manifolds

For which values of $c$ are the following sets become smooth manifolds 1) $\{(x,y)\in\mathbb{R^2} \mid x^3+xy+y^3=c \}$ 2) $\{(x_1,x_2,x_3,x_4)\in\mathbb{R^4} \mid x_1^2+x_2^2-x_3^2-x_4^2=0, x_1+...
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29 views

Deriving a curvature tensor for a special connection

May be $e(x)$ a frame vector at the point $x \in M$ for a manifold $M$. Given the following connection equation: $d_Xe+e_{|X_+}-e_{|X_-}= d_Xe + J_Xe = 0$. Here, $d_X$ denotes the directional ...
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29 views

Showing tractrix revolution is isometric to hyperbolic plane

I have a tractrix defined (in the xy plane) by the fact that the line segment formed by a tangent to the curve meeting the x axis has length 1. Given that the tangent meets the x axis at an angle $\...
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24 views

Constructing an immersion of a curve with certain properties

Consider the map $\gamma : \mathbb R \to \mathbb R^2$ given by $(t^2, t^3)$. Let $Gr(1,T\mathbb R^2) := \bigcup_{x \in \mathbb R^2} Gr(1,T_x \mathbb R^2)$ where $Gr(1,T_x \mathbb R^2)$ is the ...
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29 views

Can a bubble be modelled convincingly using a ricci flow?

My knowledge of this type of geometry leaves a lot to be desired, but I have heard about the Ricci flow. My understanding of the Ricci flow is that it models a "velocity" for each part of a shape as ...
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52 views

Interior Derivative and Contraction: Kobayashi and Nomizu.

In Kobayashi & Nomizu, the interior derivative of an r-form is defined as $\iota_X \omega = C(X \otimes \omega)$, where $C$ is the contraction associated with the pair $(1,1)$ and $\omega$ is ...
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27 views

A question on vector components with respect to an embedding on a manifold

I've been working my way through Nakahara's book "Geometry, topology and physics" and I'm currently reading through chapter 7 (on Riemannian geometry). In the 2nd section of the chapter he gives a ...
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55 views

Gauss map of a product

It is probably a stupic question, but does anybody know if the degree of the gauss map of the cartesian product of two manifolds is the product of the degrees of the gauss maps of each of them? Thank ...
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62 views

$SO(3)$ vs 3-Torus

$SO(3)$ and 3-Torus both can be viewed via rotations for a rigid body. They are not diffeomorphic. $SO(3)$ can be decomposed into three axial rotations. Could I think the reason they are not ...
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45 views

Prerequsites for working through the 2nd half of Gradient flows in metric spaces and in the spaces of probability measures

I apologize in advance if this question is too general, that is, not a request for a specific reference, but more of a request for a road map, perhaps from someone that knows the material and, in ...
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61 views

Lagrangian manifolds: basic standard theory

It is the first time that I start to learn about Lagrangian manifolds so I would like some suggestion about book and article to read. Due to the fact that is the first time, I need a book with the ...
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88 views

Change of basis formula proof

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let $M$ be an $n$-dimensional manifold and let $(U,\phi)$ and $(V,\psi)$ be two overlapping ...
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121 views

Pullback map distributes over wedge product (proof)

To prove that the pullback map distributes with the wedge product is it first best to prove that it distributes over the tensor product and then use the relation $$dx^{\mu_{1}}\wedge\cdots\wedge dx^{\...
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68 views

How to compute the unit outer normal at the point in a curve?

Given a smooth closed curve $f(x,y)=0$, How to compute the unit outer normal at each point $(x_{0},y_{0})$ in the curve?
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33 views

How do I show $f_0 \sim f_1 \Rightarrow f_0 + g \sim f_1 + g$

Consider two smoothly homotopic maps $f_1,f_2:M \to S^1$ from a compact smooth $n$-manifold $M$ to the unit circle. How do I show $$f_0 \sim f_1 \Rightarrow f_0 + g \sim f_1 + g$$ for all $g:M \to ...
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37 views

Isometry algebra implication from Riemannian covering

I really wish that, if $\pi:(M,\mathrm{g})\twoheadrightarrow(N,\mathrm{h})$ is a Riemannian covering, then $\mathfrak{i}(N,\mathrm{h})\leq\mathfrak{i}(M,\mathrm{g})$, where $\mathfrak{i}(M,\mathrm{g})$...
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52 views

A problem possibly about the Whitney Embedding Theorem.

Problem. (1) Suppose $F:S^1 \to \mathbb{R}^3$ is a smooth immersion (i.e. $dF$ is nowhere zero). Prove that there is a unit vector $\mathbf v$ in $\mathbb{R}^3$ such that $\pi_{\mathbf v} \circ F : ...
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52 views

Geometry of Curves

I found this question in question paper of Geometry of Curves and surfaces from Leeds University. Can anyone help me how I solve it.
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39 views

Envelope of a family of lines. When does it exist?

Given a family of lines $(\gamma_t): a(t)x+b(t)y+c(t)=0,\ t\in I$ (interval in $\mathbb{R}$), where $ a,b,c:I\to\mathbb{R},\ a^2(t)+b^2(t)\neq 0,\ \forall t\in I$ , what conditions should we impose ...
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41 views

Example of locally symmetric spaces

A locally symmetric manifold is a manifold with parallel curvature tensor $\nabla R=0$. Can you give an example except spheres, projective spaces and hyperbolic spaces?
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Proving that $\phi$ is orthogonal to the harmonic forms given $\int\phi \;d\mathrm{vol}$.

I want to prove that given a connected closed (compact without boundary) oriented Riemmanian manifold $(M,g)$, the condition $$\int_M \varphi \;\mathrm{d}\mathrm{vol}_g=0$$ implies that $\int \varphi ...
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45 views

Unknown proof Lie Algebra

I have a calculation where I do not know what it actually shows. I think it tells me that for right invariant vector fields, the commutator is again right invariant. Maybe somebody here could help me ...
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45 views

Dimension of Conjugacy class in $SU(n)$

Consider $D \in SU(n)$ ($n$ a multiple of 4), a diagonal matrix with values $\pm 1$ on the diagonal and with trace 0 (only possible for $n$ a multiple of 4). There are $n \choose n/2$ such matrices. ...
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39 views

Integral Over the N-Sphere in the framework of chains:

Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem: \begin{...
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114 views

Singer & Thorpe Theorem on the curvature of 4-dimensional Einstein spaces

I recently asked a question here about the paper "The curvature of 4-dimensional Einstein spaces." I got stuck again with the last theorem (2.2), where I get completely lost. They start the proof by ...
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51 views

Topological properties of regular and critical points and values

Let $f\colon M\rightarrow N$ be a smooth map between smooth manifolds. Consider the following two statements, the second one under the assumption The set of regular points of $f$ are open in $M$, ...
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56 views

Are $\mathbb{CP}^{n}$ and $\mathbb{RP}^{2n}$ diffeomorphic?

I understand that they are homeomorphic but couldn't find a proof that they are diffeomorphic. If they are diffeomorphic and if the proof is simple enough, I would imagine it would look like the ...
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62 views

intersection of normal lines converge to a point

I have some difficulty working this out. Let $\alpha: I\rightarrow R^2$ be a regular parametrized plane curve(arbitrarily parameter), and define n=n(t) and k=k(t). Assume k(t) is not 0 for $t\in I$. ...
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50 views

Basic question: Riemannian Curvature is nondegenerate

$R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}$ is the curvature with respect to the Levi-Civita connection $\nabla$ of a metric $g$ on a manifold $M$. Define the Riemann curvature ...
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29 views

How can I get this new Gaussion curvature and mean curvature?

Let $M$ be a surface in $\mathbb{R}^3$ oriented by $\vec n$. For a fixed number $\varepsilon\neq0$, let $F:M\to\mathbb{R}^3$ be a mapping such that $F(p)=p+\varepsilon\vec n(p)$ and denote $\overline{...
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39 views

Imagening the Thurston geometries

I can (more of less) imagine how it would look if space was Euclidean, spherical of hyperbolic. But there are 8 Thurston geometries see https://en.wikipedia.org/wiki/Geometrization_conjecture how ...
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51 views

How to use chain rule on it?

Let $M$ be a surface in $\mathbb{R}^3$ oriented by $\vec n$. For a fixed number $\varepsilon\neq0$, let $F:M\to\mathbb{R}^3$ be a mapping such that $F(p)=p+\varepsilon\vec n(p)$ and denote $\overline{...
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34 views

When are geodesically generated surfaces everywhere spacelike?

Suppose that $\langle M, g\rangle$ is a Lorentzian manifold, and that $\xi$ is a timelike vector in $T_pM$, at some point $p \in M$. Let $S$ be a surface consisting of points that lie on some ...
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29 views

Glueing smooth functions give a smooth function if reparametrized

Given $\mathbf{r}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$ be a $C^{1}$ application, with $\mathbf{r}'(t)=(0,0)\Longleftrightarrow\ t=t_0$, and $$\mathbf{r}(t)=\begin{cases} \mathbf{r}_1 (t)...
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49 views

1-form integration

Let $\alpha:[-1,1]\rightarrow R^2$ be the curve segment given by $\alpha=(t,t^2)$. If $\phi=v^2du+2uvdv$, (the fist component of $R^2$ is $u$ and the second one is $v$) I have $$\int_\alpha \phi=\...
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46 views

Write the vector field such that its integral curves are the meridians of $S^2$

I have problems in writing down a vector field $X$ on the sphere $S^2$ such that the integral curves of $X$ are the meridians of $S^2$. I proceed in this way. I consider this parametrisation of $S^...
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57 views

What's the name of this theorem?

If $g: \mathbb R \to \mathbb R^n$ issmooth function and $g^{(i)}(t)=0$ for $1\le i \le k-1$ and $g^{(k)}(t) \neq 0$ then there exists a smooth map $f: \mathbb R \to \mathbb R^n$ such that $g(x) = (x-...
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28 views

Local conformal coordinates on a surface

Let $\mathcal{M}\subset\mathbb{R}^3$ be a smooth enough regular surface. We want to show that around a point $p\in\mathcal{M}$ there is a neighborhood about $p$ in $\mathcal{M}$ which is parametrized ...
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34 views

Line Elements for $n$-dimensional hyperspheres

I'm currently in the process of deriving the components of the Riemann Curvature Tensor for a 3-sphere using the Cartan Equations. The line element I'm starting with is: $$ ds^2 = \frac{dr^2}{1-r^2/R^...
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166 views

The inverse image of any regular value is a submanifold

Let $f:M^m\subset\mathbb R^k\to N^n$ be a smooth map between manifolds with $m>n$. Let $y\in N$ be a regular value for $f$. Let $x\in f^{-1}(y)$ and $L:\mathbb R^k\to\mathbb R^{m-n}$ be a linear ...