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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Gaussian Curvature

Can anyone see the connection between the Gaussian curvature of the ellipsoid $x^2+y^2+az^2-1=0$ where $a>0$ and the integral $\int_0^1  {1\over (1+(a-1)w^2)^{3\over 2}}dw$? I am guessing ...
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47 views

Characterizing surfaces in $R^3$ in which every point is an umbilic point

How can it be shown that the only such surfaces are spheres or planes?
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51 views

Function seperating points

If $M$ is a Hausdorff $n$-manifold (without further assumption like paracompactness), given $x,y$ in $M$ is there a smooth function $f$ such that $f(x) \ne f(y)$?
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300 views

Deriving Euler-Lagrange equation and natural boundary conditions for two-phase piecewise $H^1$ Mumford-Shah model

Let $\Omega \subset \mathbb{R}^2$ be a bounded open domain. Derive the Euler-Lagrange equations and the natural boundary conditions for the two-phase piecewise $H^1$ Mumford- Shah model: $$J(\varphi, ...
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123 views

First-order derivatives in differential forms calculus

Let $d$ denote the Cartan differential, and let $\delta$ denote the codifferential. The underlying domain is not important for what follows. The canonical generalization of the Laplace-operator ...
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58 views

Cartan immersion of symmetric spaces.

I got stuck on some fact concerning the Cartan immersion of symmetric spaces: Let $M$ be a Riemann symmetric space associated to the Lie group $G$ with involutive automorphism $\sigma$. Let $G$ act ...
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119 views

Help on Einstein Summation

I am not sure how to interpret the following expression with regard to the Einstein summation convention \begin{equation} g^{ab}(\partial_c \Gamma^c_{ab} - \partial_b \Gamma^c_{ac}) \end{equation} ...
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97 views

Distinct metrics on a manifold

I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics ...
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673 views

Relationships between curvature, torsion, unit tangent vector, and binormal vector of a curve

Homework has already been collected and graded (but no explanation given) for these problems. I'm curious how to approach the problem. Assume that the vector space we're in is $\Re^{3}$. Prove that ...
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97 views

Another time on jets and composition

Suppose we have four smooth maps between smoot manifolds: $$f: M \rightarrow X$$ $$g: X \rightarrow N$$ $$h: M \rightarrow Y$$ $$i: Y \rightarrow N$$ an the equation on compositions of jets $$j_m(g ...
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125 views

Stereographic projections and cross-ratios

Would anybody shed some light on question 2.11 in Wilson's Curved Spaces? The numbers $p,q\in \hat{\mathbb{C}}$ are stereographic projections of points $P,Q$ on the unit sphere. The spherical ...
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63 views

Complex cones and a foliation of $\mathbb{P}^3$

1 Take the 3-dimensional complex projective space $\mathbb{P}^3$. Consider the action of the group $SU(2)\times SU(2)$. I have read in physics related articles that these group gives a singular ...
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167 views

hermitian distance functions and geometry in complex space

If we have fixed a hermitian positively definite form $h(.,.)$ in complex space $C^n$ and an analytic submanifold $M$ in $C^n$, then we may fix a point outside of $M$, say $P$, and consider distance ...
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246 views

Deriving an expression for minimum arc length along a 3D surface between any two points.

Consider a 3D surface, defined by the function $z = f(x, y)$. Assuming the surface is differentiable (no kinks), is there a function that expresses the minimum arc length traced along the surface ...
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54 views

Smooth mapping $v \colon [0,1] \to S^{n-1}$

I have a smooth mapping $v \colon [0,1] \to S^{n-1}$ such that for any $u \in S^{n-1}$ exists $t \in [0,1]: v(t)\cdot u = 0$ and $n \geq 3$. So a have an assumption that such a mapping $v(\cdot)$ ...
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186 views

Local theory of curves using Taylor expansion

Consider a $C^m$ curve $\gamma$ in $\mathbb{R}^3$ (with $m\geq 3$). Locally, at $0$ for convenience, one can express the curve as $$ \gamma(s) = \gamma(0)+sT+(s^2/2!)kN+(s^3/3!)(\dot{k}N-k^2T+k\tau ...
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74 views

Approximation in the Plane of Constant Curvature

In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve. Now, The ...
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121 views

Vector equation and curvature

Vector equation $r(t)=2\cos(t)\mathbf i + 3\sin(t)\mathbf j\ \ (0 \le t \le 2\pi)$ represents ellipse. I need to find curvature of this ellipse on endpoints of x and y axis that are given with ...
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95 views

The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
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67 views

derivatives of coordinates on a riemann surface

Let $X$ be a compact connected Riemann surface of genus $g>0$. Let $(\omega_1,\ldots,\omega_g)$ be a basis for $H^0(X,\Omega^1)$. Let $x$ be a point in $X$ and let $z:U\longrightarrow B(0,1)$ be a ...
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173 views

Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
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115 views

Decomposition of linear partial differential operators

I was wondering about the following: Let $M$ be a smooth, second-countable (possibly noncompact) manifold and let $E$ and $F$ be smooth vector bundles over $M$. Can every smooth linear partial ...
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929 views

How do we prove that a sphere maximizes the volume enclosed among all simple closed surfaces of given surface area?

How do we prove that among all closed surfaces with a given surface area, the sphere is the one that encloses the largest volume, and not do it by cases? so far I've tried is that I know the formula ...
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304 views

Question on Stokes' Theorem

Suppose $M$ is a smooth manifold and $f$ is a real valued smooth function on $M$. Set $N:=f^{-1}([0,1])$ and suppose $N$ is a compact submanifold of $M$. Let $\mu$ be a volume form on $M$ and $v$ a ...
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366 views

a question about the geodesic circle

How to show that the geodesic circles have constant geodesic curvature on a surface of constant curvature? Thanks
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157 views

costructing a diffeomorphism

Let $i:N\to M$ be a smooth embedding, $\pi:E\to N$ a vector bundle and $s_0:N\to E$ is its zero section. I have an open neighborhood $U$ of $s_0(N)$ in $E$, and $f:U\to M$ is a smooth map such that ...
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147 views

A particular pulling back and lifting of metric

Let $\Sigma$ be a $n-1$ dimensional space-like submanifold of a $n+1$ dimensional space-time $(V,g)$ and let $x \in \Sigma$. Then $(T_x \Sigma)^\perp$ is of dimension $2$ and is time-like. Such a ...
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257 views

A particular method of pulling back a metric on a submanifold

Let $S$ be a $(n-1)$-submanifold of a $n$-manifold $M$ and that be a submanifold of $(n+1)$-manifold V. (All of the manifolds are assumed orientable) Let $V$ carry a metric of signature $(1,n)$. Using ...
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13 views

Paramatrizes curve with constant speed

Show that if $\alpha : I \rightarrow \Re^{n+1}$ is a parametrised curve with constant speed then $\alpha(t) \perp \frac{d}{dt} \alpha(t)$ for all $t\epsilon I$.
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46 views
+50

Isometric and conformal map

We defined conformal and isometric maps for surfaces $f,g: \Omega \subset \mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3$. Under a reparametrization of $f$ I understand a diffeomorphism $\Phi : M ...
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21 views

Use Frenet Frame and Pythagorean Theorem.

Suppose we have a curve $c(t)$ where $t$ goes from $a$ to $b$. $c$ has positive curvature and a frenet frame(TNB). Choose $\rho > 0$ and small. and define: $f(t) = c(t) +\rho B(t)$ $g(t) = c(t) ...
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22 views

Are there two kinds of Christoffel symbols?

I am struggling to understand Christoffel symbols. Part of my confusion is that there are two kinds. So I mix up which properties belong to each and end up learning about neither. Can someone define ...
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25 views

Transversality of leaves to the spheres .

Consider a form in the complex plane such that its linear part is $\omega_0=\lambda_1xdy-\lambda_2ydx$ in the Poincare domain: $\lambda_1\lambda_2 \ne 0$ and $\lambda_1/\lambda_2 \notin \mathbb{R}^-$. ...
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31 views

Geometrical definition of the first fundamental form?

I am looking for a geometrical interpretation of the first fundamental form. I would guess it is something like a small line element that sits in the surface which then integrated over will give the ...
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41 views

Vanishing Christoffels symbols

Under what conditions does there exist a parametrization of a surface, for which the Christoffel symbols are zero. I heard that has something to do with "flat connection". I would like to see proofs.
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31 views

Computation of the first fundamental form of ruled surfaces

It is possible to prove that ruled surfaces can be parametrized as follows: $\overrightarrow{X}(t,u)=\overrightarrow{\beta}(t)+u \overrightarrow{w}(t)$ where $\|\overrightarrow{w}(t)\|^{2}=1$ and ...
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51 views

Vectors in tangent space to a manifold independent of coordinates

In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator $X$ acting on some function ...
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21 views

Differentiability of a function on a manifold is independent of the coordinate chart

I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a ...
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51 views

Integral in Gauß Bonnet theorem

I just read a text about the Gauß Bonnet theorem. If I have a function $f:\Omega \rightarrow M$ defining a two-dimensional manifold $M$ with a boundary that is parametrized by a curve $c: I ...
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17 views

Assymptotic direction

Me and my classmates are interested in a visual description of an asymptotic direction at a point of a surface. The normal curvature in an assymptotic direction at a point is zero. And a curve on a ...
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48 views

Charts of $\wedge T^*M$ and $\wedge TM$.

Let $M$ a differentiable manifold and consider $\wedge T^*M$. If $(U, \phi)$ is a chart of $M$ with coordinate functions $(x_1, \dots, x_n)$, then $\{ \frac{\partial }{\partial x_i}\}_i$ and $\{ ...
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23 views

Time Derivative of the integral over a singular k-cube

I am stuck on this question, and was wondering if someone could provide a hint of where to start? I can't see the first step.
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41 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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32 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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23 views

Arc length parametrization of parameter curves of the sphere

I would like to find a parametrization of (part of) the sphere where the parameter lines are arc length parametrized. The reason is that I was asked to show that if the parameterlines of a surface ...
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29 views

Fundamental group of cusp of a negatively curved manifold

Let $M$ be a complete, noncompact Riemannian manifold with finite volume and whose sectional curvatures vary within the interval $[a,b]$, $-1\leq a<b<0$. It is known that such manifold has ends ...
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17 views

Rotationally non-symmetric Sine Gordon application

Has the intrinsic Sine-Gordon equation been ever used to define asymptotic lines on constant negative Gaussian curvature surfaces of Kuen, Breather or other rotationally non-symmetric surfaces ? ...
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43 views

Riemannian Connection

How can we see for the Riemannian connection, connection 1-form with its first index lowered $\omega_{ab}=\delta_{ac}{\omega^c}_b$ is antisymmetric in a, b, i.e. $\omega_{ab}=-\omega_{ba}$. Thanks.
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32 views

One parameter subgroups on Lie groups and Riemannian metric

I read that geodesics of a bi-invariant metric on a compact Lie group are the one parameter subgroups. In a general Lie group, is it possible to create a Riemannian metric by transporting the ...
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39 views

How does an affine connection permit differentiation of vector fields?

As I understand it, one primary use of affine connections is to "connect" tangent spaces. Suppose I take a velocity vector $\dot{\gamma}(t_0)$ on a curve and at some point $\dot{\gamma}(t)$ also on ...