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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Sectional curvature in 3-dimensions

I wonder how to compute the sectional curvature of 3-dimensional objects eg. unit ball, $H=\{(x_{0},x_{1},x_{2},x_{3})\in \mathbb{R}^{4}:x_{0}^{2}-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})=1$ and ...
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Does there exists known special cases of a zero Riemann tensor for 3D metrics?

In two dimensions, if one has a flat metric $g_{ab}$, then one can transform $g_{ab}$ to another flat metric $h_{ab}=e^{2\varphi}g_{ab}$, when $\nabla^2 \varphi =0$ and the Riemann tensor remains ...
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74 views

Show $f:\mathbb{R}^n \to \mathbb{R}^m$, $n>m$ can't be 1-1

Problem 2-37 on p. 39 of Spivak's Calculus on Manifolds asks Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a continuously differentiable function. Show that $f$ is not 1-1. (Hint: If, for example, ...
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24 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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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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32 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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50 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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34 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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52 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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49 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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43 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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34 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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26 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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32 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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42 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 ...
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29 views

Does existance of a Killing spinor imply existance of a Killing vector?

I am wondering about the relationship between Killing spinors and Killing vector fields. In the nlab entry for Killing spinors the quote "Pairing two covariant constant spinors to a vector yields a ...
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26 views

prove that $α_{:k}=0$ and $β_{:k}=0$?

Let $α$ be a Riemannian metric and $β$ a one form. The question is: if $F= α+β$ then find $F_{:k}$ ? Also prove that $α_{:k}=0$ and $β_{:k}=0$?
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3D Involute of constant geodesic/ Gauss curvatures

Is my following proposition correct? Unwinding a taut surface-contacting geodesic thread to trace an involute with constant geodesic curvature $ k_{g}$ is possible only on $ \mathbb R^2 $ ...
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43 views

Local expression of a differential form

During the course, we have defined differential forms as maps $$ \omega : D(M) \times \cdots \times D(M) \rightarrow \mathcal{C}^\infty(M), $$ where $D(M)$ are the global derivates of the manifold ...
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37 views

Limits of visualizing $p$-forms?

On page 90 of Gravitation, Misner Thorne and Wheeler state the following: Stacks of surfaces, individually or intersecting to make "honeycombs", "egg crates", and other structures ("differential ...
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36 views

Does the classifying map of a fibre bundle only depend on the transition functions?

Does the classifying map of a fibre bundle only depend on the transition functions? Precisely, Let $\xi$ and $\eta$ be two fibre bundles over $B$, whose transition functions are same, both of their ...
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26 views

Right Veering Property of elements in MCG(S)

Let h be an element of MCG(S), the mapping class group of a surface S. I was going over : I was going over :Geometric Intuition for "Right-Veering" Property of $f$ in MCG(S)? Where a p.e ...
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35 views

normal exponential map on Riemannian manifolds

where can I find information about normal exponential map on Riemannian manifolds? please introduce some books.
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53 views

Show that $\Psi_{t*}\mathbb{X}-\Phi_{t*}\mathbb{X}=\mathbb{X}-\mathbb{Y} $

Let $\mathbb{X},\mathbb{Y}$ be vector fields and let $\Phi_t$ denote the flow of $\mathbb{X}$. Given that $\displaystyle \frac{\partial}{\partial t}\Phi_{t*}\mathbb{Y} ...
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24 views

Find a complex-valued $g(u,v)$ such that $L_\mathbb{Y}g=img$

Let $F$ be a diffeomorphism between open $U$ and $V$ in $\mathbb{R}^n$. Let $\mathbb{Y}$ be a vector field on $V$ and $\omega$ a $k$-form on $V$. Given the identity ...
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40 views

Show that $\int_b d\omega=0$ where $b(s,t)=\Phi_s(c(t)) $

Let $c:[0,1]^k \rightarrow \mathbb{R}^n$; $t \mapsto c(t)$ be k-cell with $k < n$. Let $\mathbb{Y}$ denote a vector field on $\mathbb{R}^n$ with flow $\Psi_s$. Define a $(k+1)$-cell $b:[0,1]^{k+1} ...
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21 views

Is a compacthypersurface $S \subset \mathbb{R}^n$ a “conformally compact Einstein manifold”?

If $S$ is a hypersurface in $\mathbb{R}^n$ (like a sphere) then is $S$ a "conformally compact Einstein manifold"? It's a compact manifold. According to the wiki, it is a conformal manifold if it is ...
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17 views

Looking for a relationship between two push-forward maps

In a push-forward question I was to compute $(F_*\mathbb{X})(F(u,v))$, the next part is to calculate $(F_*\mathbb{X})(x,y)$. I have computed the first part and am able to compute the second part, ...
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32 views

on isometric group

Let $G_n$ be a Lie group, $g$ denotes the left invariant Riemannian metric on $G$. I want to ask for help that how to prove this conclusion: if all principal Ricci curvature of $(G_n, g)$ are ...
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55 views

Example of implicit function theorem

I'm trying to understand implicit function theorem, so I came up with an example myself: Let's define: $f:{\mathbb R}^2\times{\mathbb R}^2\to{\mathbb R}^2, (a,b,c,d)\to (ad,bc)$. Let ...
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44 views

The level set of Lipschitz functions

Suppose $u$: $R^N\to R$ is lipschitz, then do we have a.e. level set of $u$ has Lipschitz boundary? Is this anything to do with Sard theorem? Sard theorem states that a.e. Level set of smooth ...
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17 views

Finding parallels surface of revolution

$$X(u,v)=(f(u)\cos(v),f(u)\sin(v),g(u))$$ where $v$ is the rotation angle around $z$ axis. What is the parallel of this surface of revolution?
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22 views

Every point in the codomain is a regular point

Let $0<r<1$ and define $f:\mathbb R^3\to\mathbb R$ by $$f(x,y,z)=(x^2+y^2+r^2-z^2-1)^2-4(x^2+y^2)(r^2-z^2).$$ Let's denote $x^2+y^2=a,r^2-z^2=b$. I don't know if this is allowed, but I just ...
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34 views

Completion of hyperbolic 3-manifold (Thurston's lecture note)

I need help with a part of Thurston's lecture note. p.55~56 in the 4th lecture note states that when obtaining hyperbolic manifold by gluing polyhedra having some ideal vertices, the set of ...
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16 views

Path of constant relative gradient to a cone.

The equiangular or logarithmic spiral has the property that the angle between any tangent and the radial line is a constant. I am looking for a curve with the same property with respect to a conical ...
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23 views

Explicit formula for the (n-2)th derivative of the Jacobi equation

The $n-2$ order derivative of the Jacobi equation is given by: $$\frac{D^n}{dt^n} V_i+\sum\limits_{l=0}^{n-2} \binom{n}{k} (\nabla_{\gamma '}^{(n-2-l)}R)(\gamma ' ,\nabla_{\gamma '}^{(l)} V_i)\gamma ...
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39 views

Pull-back of a one-form on a sphere.

Let $\iota: S^2 \to \mathbb{R}^3$ be the inclusion map and choose a chart $(U,f)$ on $S^2$, where $U=\{(x,y,z)\in \mathbb{R}^3: z>0\}$ and $$f: U \to \mathbb{R}^2,$$ $$ (x,y,z)\mapsto (x,y). $$ I ...
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56 views

What geometric shape is “the perfect milkshake container”?

A sphere is the best shape for a snowball if you want to maximize the amount of time before the snowball melts. This is because the ratio of the surface area divided by the volume is the smallest. ...
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13 views

Surfaces swept out by trihedron vectors

Surfaces swept out by unit tangent of a curve on a surface is developable. Are normal and bi-normal swept out surfaces also developable?
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27 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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23 views

Oval surface with $ K \ge 1 $ which is unit sphere

Let $ S$ oval surface with $ K \ge 1$ . If there is exist an open unit sphere interior of $ S$, then $S$ is unit sphere...Can anyone give me an idea of the solution...thanks in advance...
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24 views

Define the concept of the Shape Operator and Fundamental Forms

I am confused about the relationship between three concepts: shape operator, first fundamental form, second fundamental form. I would like someone to provide me with a basic definition of these terms ...