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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tubular neighborhoods - problem

Consider $\alpha:[0,L]\to \mathbb R^3$ a smooth regular simple closed curve, arc-length parametrized. Denote its trace by $\gamma$. Let $\epsilon >0$ be a real number and $N_\epsilon (\alpha)$ the ...
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128 views

Given Constant Ratio of Torsion to Curvature, Show Tangent times Constant Vector is Constant

Let $r(t)$ be a unit speed curve such that for all $t$, $\frac{\tau(t)}{\kappa(t)}=\cot(\theta)$ for some $0 < \theta < \pi$. Show that there is a constant vector $a$ satisfying $T(t) * a = ...
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39 views

PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...
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22 views

How do I figure out the grid references necessary to create a circular 1 mile radius around one location?

How do I figure out the grid references necessary to create a circular 1 mile radius around one location? For instance: Grid Reference: SO 47904 87484 X (Eastings): 347904 Y (Northings): 287484 ...
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92 views

Surface of revolution

This problem is from Dubrovin's Modern Geometry (Problem 8, Exercise 8.4). Show that the only surfaces of revolution with zero mean curvature are the plane and the catenoid (which is the surface ...
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29 views

Help modeling 3d vector field

I'd like some help in finding the correct mathematical description of the stuff below: A sheet is deformed by a mass on it (like in one of those pictures showing the effects of General Relativity). ...
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259 views

Chain rule with covariant derivative

Let $\mathcal{M}$ be a $n$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. Consider the following function: $$\tilde{F}(v) = \operatorname{d exp}^{-1}_{p} ...
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48 views

Boundaries of a cycloid

Supposing the Catenoid is parametrized by $x(u,v)=(acosh(u)cos(v),acosh(u)sin(v),au)$,where $u$ is a real number and $0<=R<2\pi$ and $a>0$ is fixed. Given the pair of parallel circles ...
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63 views

Nonvanishing vector fields on $\mathbf{S}^2$ and bases for $T_p\mathbf{S}^2$.

A vector field on a manifold is a (continuous, differentiable) map $X: M \to TM$ such that $X(p) \in T_pM$ for each $p \in M$. The tangent space $T_pM$ has a basis. Take $M = \mathbf{S}^2$. For each ...
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40 views

Show that x defines a coordinate neighborhood on the upper sheet of the circular cone C

Consider the map $x : U = (0, 2π) \times (0,\infty)\to{\mathbb R}^3$, $(\theta, v) \mapsto (v\cos\theta, v\sin\theta, v).$ a) Show that $x$ defines a coordinate neighborhood on the upper sheet ...
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56 views

Is the $\varepsilon$-neighbourhood theorem used in proving Homotopic transverse extension?

In Guillemin & Pollack page 71 I can't see where "For compact mfld Y,the map $\pi:Y^{\varepsilon}\to Y$ is a submersion" is used to show: "If for $f:M\to N$, closed subset $C\subset M$, closed ...
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63 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
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64 views

Proving a curve of a logarithimic spiral is 1/s cot(theta)

A unit-speed plane curve $\gamma$ has the property that its tangent vector $t(s)$ makes a fixed angle $\theta$ with $\gamma(s)$ for all $s$. Show that: (i) If $\theta = 0$, then $\gamma$ is part of a ...
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34 views

If 2 Equations with different variables are equal, then they are constant?

Let $\phi:\mathbb R\rightarrow\mathbb R$, $\psi:\mathbb R\rightarrow\mathbb R$ be $C^{\infty}$ maps, and $f:\mathbb R^2\rightarrow\mathbb R^3$ by $$f(u,v):(u,v,\phi(u)+\psi(v))$$ and set ...
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53 views

Getting started with contact bundles

I'm currently reading William Burke's book Applied Differential Geometry and he uses a lot in the development of Lagrangian Mechanics the notion of a Contact Bundle. He does explain intuitively what ...
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67 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
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89 views

Consequences of tubular neighbourhood theorem

Consider an oriented manifold without boundary S embedded in an oriented manifold M. The tubular neighbourhood theorem says that there is a neighbourhood T of S in M which is diffeomorphic to ...
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56 views

Laplace-Beltrami operator for Kahler 2-form

Laplace-Beltrami operator for Kahler 2-form: $$\triangle\Omega(X,Y)=d\delta\Omega(X,Y)+\delta d\Omega(X,Y)$$ We know that $$\delta d\Omega(X,Y)=-\sum_{k}(\nabla_{e_{k}}d\Omega(e_{k},X,Y))$$ where ...
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51 views

Chains and cochains: integer versus real coefficients

Let a real, smooth manifold $M$ be given. For each non-negative integer $k$, let a singular $k$-cube on $M$ be a continuous mapping $c:[0,1]\to M$. Let $C_k(M,\mathbb Z)$ denote the set of formal ...
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48 views

Wedge product of a one- form and a Kähler form

Let $x$, $y$, $v$, $w$ be coordinates on $R^{4}$ and $g$ be the Riemannian metric whose matrix with respect to these coordinates is $$g=\left ( \begin{array} {cccc} 1 & 0 & -kx & 0\\ 0 ...
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58 views

Tanget space to manifold via curves without map

I define tangent space T to differentiable manifold, in point p, via equivalence class of curves. The condition for this equivalence is $(\varphi\circ\gamma_1)'(0)=(\varphi\circ\gamma_2)'(0)$ for some ...
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33 views

Proving a global property of the connection from a property of local connection matrices

Let $D$ be an $m$-dimensional distribution on an $n$-dimensional manifold $M$. Let $U\subset M$ be an arbitrary open subset such that on $U$ we can define vector fields $e_i$ such that for each $x\in ...
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49 views

Tensor vector bundle construction

$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes ...
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55 views

Prove that a canonical bundle is trivial

Consider a function $f \in C^{\infty}(\mathbb{R}^n)$, $y \in Reg(f), M=f^{-1}(y)$. Prove that the canonical bundle of M is trivial. I have an hint but I don't know how to use it: consider the open ...
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73 views

Structure equations on the 3-sphere

On the 3-sphere I have found the vector fields $X_1=(-x_2,x_1,-x_4,x_3)$, $X_2=(-x_3,x_4,x_1,-x_2)$, $X_3=(-x_4,-x_3,x_2,x_1)$, in the basis $\left\{\frac{\partial}{\partial ...
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83 views

How to prove that the composite function is smooth

Let $f:M \to N$ and $g:N \to K$ be smooth functions, where $M,N$ and $K$ are smooth manifolds. How to prove that the composite function $g \circ f$ is smooth, noting that the Chain Rule only applies ...
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69 views

construction of a special series of functions

Here is the problem: Let $A$ be the set of positive integers greater than 1. For each $L\in A$, we want to construct a smooth function $f_L$ with compact support such that ...
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44 views

Some properties in differential geometry of curves and surfaces

Let $\beta, \alpha$ be curves in $\mathbb{R}^3$ parametrized by arc length. Suppose $\beta$ is obtained by rotating $\alpha.$ Let $t_{\alpha}, n_{\alpha}, b_{\alpha}$ (resp. $t_{\beta}, n_{\beta}, ...
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25 views

Irreducible Decompositions of the Space of Sections of an Equivariant Vector Bundle

Let $G$ be a compact semi-simple Lie group, and $X = G/H$ a homogeneous space of $G$. If $V$ is a vector bundle over $X$, then is it true that $\Gamma^\infty(V)$ always has a decomposition into finite ...
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56 views

Vector fields on homogeneous space $G/H$

I am trying to understand why the vector fields on $G/H$ are maps $X:G\rightarrow \frak{g}/\frak{h}$ satisfying $X(rh)=Ad^{-1}(h)X(r),\,\, h\in H.$ Any hint would be greatly appreciated!
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157 views

Some Advice on My Undergraduate Paper

My teacher wants me to read something about "Differential Geometry in $R^3$" and choose a topic as a paper. Now I have finished these books. And I am interested in some topics below: $(1)$ ...
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55 views

Doubt on the definition of topological manifold

I need some clarifications on topological manifolds. My professor has defined them as second countable, connected, Hausdorff topological spaces which are locally homeomorphic to $\mathbb{R}^n_+$, ...
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149 views

The Fubini-Study metric and its associated form.

Assume $ds^2$ is Fubini-Study metric, and $\omega = -\frac{1}{2}Im \{ds^2\}$ is its associated form. Let $V_n = \int_{\mathbb{C}P^n} \omega^n / n!$ the volume of $\mathbb{C}P^n$. Let $V \subset ...
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272 views

The curvature and torsion of the tangent indicatrix

Let $\alpha$ be a unit speed curve. Its tangent indicatrix $\sigma$ is defined by $\sigma(t)=T(t)$. Find torsion and curvature of $\sigma$ with respect to the torsion and curvature of $\alpha$. ...
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96 views

Expressions for exponential map and parallel transport

This came up in a paper I was perusing. The authors list three formulas which I have not been able to comprehend. Here M is supposed to be a simply connected space form. Then for the exponential map ...
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62 views

Newton polygon and asymptotic behavior near a singular point

As we know, Newton polygons could be used to determine the Puiseux series of algebraic curves (see, for example, Kirwan's Complex Algebraic Curves, chapter 7). Different branches correspond to ...
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117 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
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78 views

The relation between conformally related metrics and conformal vector fields?

Two metrics $g_{1}$ and $g_{2}$ are conformally equivalent metrics if $g_{2}=e^{2\theta}g_{1}$ A vector field $X$ is called conformal if $L_{X}g=2\theta g$ where $L_{X}$ is the Lie derivative with ...
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30 views

What can we say about the integral curve of a vector field on the warped product manifold?

Let $Z=(X,Y)$ be a vector field defined on the warped product $M×_{f}N$ where $f$ is defined on $M$. The integral curve of $X$ on $M$ is $\alpha$ and the integral curve of $Y$ on $N$ is $\beta$. I ...
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43 views

comformal surface parameterization of simply-connected Riemann surface with boundary

$M$ and $S$ are two simply-connected surfaces with boundaries $\partial M =\partial S \neq \emptyset$, respectively. Both $M$ and $S$ are disk-like and smooth enough. Here, we assume $M$ is planar ...
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67 views

Frenet equations used in curved space

I've been reading this article by M. Abramowicz. Very interesting article but I can't make sense of one detail: the author writes Eq.$(9)$ stating that we can use (that) Frenet equation to define the ...
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Orientability of Ringed Space

Differential manifold can be defined in two ways. One definition is a topological space equipped with an atlas and transition maps. Another definition is a topological space equipped with a sheaf of ...
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98 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
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41 views

Minimal immersion

Let $N$ be an $n$-dimensional manifold immersed in an $l$-dimensional manifold $L$ by immersion $\iota: N\to L$. And let $\iota$ be minimal immersion. On the other hand, let $\phi: M\to N$ be totally ...
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51 views

Smooth parametrizations of continuous curves

Consider the curve $t\to (t,|t|)$ in $\mathbb R^2$. Even if it has a cusp in $(0,0)$ i can reparametrize it with a smooth function. Take for example $$ t\mapsto \begin{cases}(\mathrm e^{-1/t},\mathrm ...
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66 views

Mixed dimension non-Euclidean geometry?

Is the following a "consistent non-Euclidean geometry"? It seems to satisfy the first 4 Euclidean postulates. Any comments? Any agreements or disagreements? Following are the additional conditions on ...
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58 views

Tangent space of Sym(n,$\mathbb{R}$)

I want to compute the tangent space of the space of symmetric real matrices Sym(n,$\mathbb{R}$) = $\{A\in GL(n,\mathbb{R})|A^t=A\}$. There are two different (and seemingly contradictory) statements ...
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78 views

Identites of Riemann curvature tensor on orthonormal frame

Suppose we consider a Riemannian manifold and a local orthonormal frame $\{Y_i\}$. I was wondering whether, for the Riemann curvature tensor $R$, there are identities with regards to expressions of ...
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47 views

Determine the number of revolutions the normal to a curve makes as it moves along a curve in three dimensions?

Given a point on a 3D curve, how many full revolutions does the normal to the curve at the point make as the point moves over the curve? Assume the point stops when it reaches the place where it ...
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44 views

immersions of spheres with handles minus disks

i am at a loss as far how exactly to immerse a sphere with genus g minus a disk into the plane. Thanks in advance