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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Check that the parametrization x(u,v)is conformal if and only if E=G and F=0.

Check that the parametrization x(u,v)is conformal if and only if E=G and F=0. I am slightly confused with what this question is asking me. Could someone please walk me through this question. I ...
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How to parallel transport a coframe field in a geodesic normal neighborhood?

From Chern: Lectures on Differential Geometry, page 147 Chern claimed that a torsion-free connection is completely determined locally by the curvature tensor. To show that he considered a geodesic ...
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55 views

A space curve is planar if and only if its torsion is everywhere 0

Can someone please explain this proof to me. I know that a circle is planar and has nonzero constant curvature, so this must be an exception, but I am a little lost on the proof. Thanks!
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30 views

Naturality of Lie derivatives

It was left to me as an exercise to show the naturality of the Lie derivative of arbitrary tensors on a smooth manifold. Is the following argument correct (it seems too easy)? Let $X$ be a vector ...
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24 views

Orthogonal transformation and vector product

I found these thing in an exercise 1.5.6 in the book Differential Geometry of curves and surfaces - Do Carmo. "Show that the vector product of 2 vectors is invariant under orthogonal transformation ...
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27 views

Finding the surface area of a parametrized surface

I was wondering how you would compute the surface area of a parameterized surface. Is there a formula or set of procedures you can follow to compute this. Say I wanted to compute the surface area of a ...
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32 views

When to take derivative with respect to distance?

I had a previous question about the divergence in spherical coordinates and using the usual formula found on wikipedia "List of formulas in Riemannian geometry" I could not get the correct form of the ...
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37 views

Norm equivalence of p-forms

Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$. $g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$. $A^p$ denotes the set of p- forms on ...
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20 views

Implicit representations of a regular surface.

Suppose that $\mathcal{S}$ is a regular surface and $f(x,y,z)=0$ is an implicit representation of this surface in a neighbourhood $V$. Can it be shown in general that at any point of ...
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15 views

Relationship between differentiation and integration of vector fields?

Let $V\in\Gamma(T\mathbb{R}^n)$ be a vector field and $\gamma:[a,b]\to \mathbb{R}^n$ a curve. Let $\nabla$ be the Euclidean connection, i.e. $\nabla_XY=XY^k\frac{\partial}{\partial x^k}$. We have a ...
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43 views

Degree of a vector field on the boundary of a punctured ball?

Suppose $v\colon \mathbb{R}^3\setminus\{x_1,\dots,x_n\}\to S^2$ is a smooth map, where all the points $x_i$ sit inside the unit ball $B^3$, and not on the boundary $S^2$. I suppose this map can be ...
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42 views

Why are parallel vector fields called parallel?

In Lee's "Riemannian Manifolds: An Introduction to Curvature" given a curve $\gamma:[a,b]\to M$ and a tangent vector $V_0\in T_{\gamma(t_0)}M$, where $t_0\in [a,b]$, there is a drawing of the parallel ...
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27 views

Real/Complex Manifolds - Transition Maps

I'm trying to understand how real/complex structure is imposed on a manifold, especially the likes of smooth manifolds. I can read the definitions and work with them, but I want to understand ...
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Computing transition map of $S^2$.

First please have a look at the cruddy diagram I have drawn. (it is at angle because my camera casts a shadow if I photograph it from above) Define the coordinate charts that map a portion of the ...
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38 views

Cohomology class with trivial restriction to a very general fiber

Let $f:X\to S$ be a flat morphism of smooth complex projective varieties. Let $s\in S$ be a very general point. Suppose that $\omega\in H^{p,p}(X)$ is a cohomology class such that $\omega|_{X_s}=0\in ...
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Proof of Euler's Theorem on differential geometry [on hold]

Proof of Euler's Theorem on Differential Geometry: K(A) = k1 Cos^(A) + k2 Sin^(A) Thank you.
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23 views

Tangent of evolute and singed curvature

This is an exercise from differential geometry textbook by Do Carmo. Let $\alpha:I\to \mathbb{R}^2$ be a regular parametrized plain curve (arbitrary parameter), define $n=n(t)$ and $k=k(t)$, where ...
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19 views

Which parts of $S$ these parametrization cover?

Show that the set $S=\{(x,y,z): z=x^2-y^2\}$ is a regular surface and check that parts a and b are parametrizations for $S$: a. $x(u,v)=(u+v, u-v, 4uv)$ with $(u, v)\in\mathbb{R}^2$ b. ...
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21 views

Orientation of surfaces

From the book: Fixing a parametrization $x(u,v)$ of a neighborhood of a point $p$ of a regular surface $S$, we determine an orientation of the tangent plane $T_p (S)$, namely, the orientation of the ...
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36 views

Is this orientation preserving or reversing?

I am confused about the definition of orientation on manifolds. Let $X=\{(x,y,0)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ and $Y=\{(x,y,1)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ be two one dimensional circles in ...
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34 views

Is a sphere really a (differentiable) manifold?

I am a beginning student in Differential Geometry. According to what I understand, the charts: $$\sigma_+^z(x,y) = (x,y, \sqrt{1 - x^2 - y^2} )$$ $$\sigma_+^x(u,v) = (\sqrt{1 - u^2 - v^2},u,v )$$ ...
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Gauss curvature K in polar coordinates

EDIT: A surface is given in Monge's form: $z=f (x,y)$ the partial derivatives of $z$ are.. $$ p = \frac{\partial z}{\partial x}, \; q = \frac{\partial z}{\partial y}, \; ...
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43 views

Inverse mapping for a simple $\mathbb{R}^3$ surface given by $(\sin u, \sin 2u, v)$.

For a domain $U=\{\, (u,v) \in \mathbb{R}^2 \mid -\pi<u<\pi,\ 0<v<1 \,\}$ we have a mapping $X \colon U \to \mathbb{R}^3$ defined by $X(u,v) = (\sin u, \sin 2u, v)$. The resulting surface ...
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Sobolev space of p-forms on a Riemannian mamifold

Let $(M,g)$ be a compact Riemannian manifold of dimension $d$. Let $(U';\varphi =x^1,\cdots, x^d)$ be a chart of M, $U\subset\subset U'$ be an open set of $U'$. $A^p(M)$ denotes the set of smooth ...
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Is it possible to rectify two linearly independent vectors by the same diffeomorphism to the first two unit vectors in $\mathbb R^n$?

Suppose we are given two vector fields $V_1$ and $V_2$ defined on $\mathbb R^n$ such that the vectors $V_1(x)$ and $V_2(x)$ are linearly independent for each $x$. Is is possible to find a ...
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Question about the existence of the flow of a vector field in $\mathbb R^3$.

Let $V_r$ be a smooth vector field defined on a sphere of radius $r$ that is always tangential to the sphere on which it is defined. Define a vector field on $\mathbb R^3$ by declaring $V(x) = ...
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80 views

Confusion regarding Riemann normal coordinates

I'm trying to understand Riemann normal coordinates. This "simple" example using the surface of a unit sphere is from http://www.maths.bris.ac.uk/~macpd/gen_rel/snotes.pdf (p26). The “north pole” ...
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39 views

Notion of a distribution as acting on tangent spaces

I'm reading a paper that uses distributions in a way I'm not entirely comfortable with. To be precise, I'm not sure what definitions the author is working with and can't find any natural way to fill ...
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23 views

Symplectic form and volume of parallelepiped

Define the canonical symplectic form $\omega$ on $\mathbb{R}^{2n}$ by $\omega(u,v)=u^TJv$, where $$J=\begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}.$$ I do not understand why the volume ...
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28 views

$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
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Geodesic formulation from surface parametrization

What differential relation f (u,v,du/dv)=0 can be used to convert parametrization of a two parameter surface X(u,v) into one parameter geodesics on surface X ?
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Evolute of a cycloid

Find the evolute of the cycloid: $x = u + \sin(u)$, $y = 1 + \cos(u)$ Hint: given is that $T = \big(\cos (u/2), − \sin(u/2)\big)$. Is it that we differentiate $x$ and $y$ and then make it ...
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49 views

the shortest path between two points and the unit sphere and the arc of the great circle

Prove that the shortest path between two points on the unit sphere is an arc of a great circle connecting them Great Circle: the equator or any circle obtained from the equator by rotating further: ...
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Uncertain about answers computing area and volume of sphere with unusual metric.

Consider the a metric in a three dimensional space given by $$ ds^{2} = \frac{dr^{2}}{1-\frac {2}{r}} + r^{2} (d\theta^{2} + sin^{2} (\theta) \: d\phi^{2}) $$ Calculate: a) The area of a sphere ...
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31 views

Riemannian geometry algebra

Is this derivation correct? $$ R^{ab}_{;a}=0 $$ $$ g_{ac}g_{bd}R^{ab}_{;a}=0 $$ $$ (g_{ac}g_{bd}R^{ab})_{;a}=0 $$ $$ R_{cd;a}=0 $$ And does that mean I now have $n^3$ equation as opposed to $n$?
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Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...
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Some problem similar to Dido's problem [duplicate]

The question is : "Let $A$ and $B$ be two fixed points in $\mathbb{R}^{2}$. Given $L>$ length of $AB$. Show that the curve $\alpha$ joining A and B, with length $L$, which together with AB forms a ...
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mean curvature of a nonlinear parabolic PDE

I want to compute the mean curvature of a reaction diffusion equations of the form $$ u_t=u_{xx}-u^2 $$ Any suggestions is appreciated!
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1answer
35 views

computing transition function of tangent bundle $S^n$

I'm just starting to learn about vector bundles, I want to compute the transition functions of the bundle $TS^n$. I started with the stereographic atlas $U_1 = S^n - \{N\}$ and $U_2 = S^n - \{S\}$ ...
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1answer
34 views

Under what condition on f is this parametrized curve regular?

Consider a parametrized curve in $\mathbb R^2$ given by $$ \gamma (t)=(f(t)\cos(t), f(t)\sin(t)) $$ where $f$ is a smooth function of $t$. Under what condition on $f$ is $\gamma$ regular? I took the ...
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48 views

Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
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Transitivity of smooth submanifolds

I was reading through Guillemin and Pollack and was having trouble verifying this for myself. Given $M \subset N$ and $N \subset P$, where $M$ is a submanifold of $N$, and $N$ a submanifold of $P$, ...
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Question surface

We consider the surface $S$ that is defined as the graph of the function $z=2x^2-y^2, x,y \in \mathbb{R}$ Find a basis of the tangent plane $T$ of the surface $S$ at the point $M=(-1,2,-1)$ Find a ...
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Relation between $\text{Hom}_{\mathsf{Alg}_{\mathbb{R}}}(\mathcal{C}^\infty(M),A) $ and $ X \otimes_\mathbb{R} A$?

This question is a little bit of a shot in the dark, but maybe someone stumbled over it before... Let $M$ be a (simply connected) smooth manifold modelled on a locally convex space $X$ over ...
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How to organize my learning in Maths?

I m working on a problem in mechanics of material which concerns about the variation of shapes. I need to understand the deformation of material. I m a civil engineering graduate. All my understanding ...
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62 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
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32 views

Hausdorf Dimension of a manifold of dimension n?

Let's say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus ...
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Problem 1, Chapter 3 - A Comprehensive Introduction to Differential Geometry - Spivak

There seems to be an error in item (d) of Problem 1, Chapter 3 of Spivak's A Comprehensive Introduction to Differential Geometry, 3rd Edition, which I transcribed below. There is something wrong with ...
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2answers
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Einstein Summation - does the following equality hold: $a_{ij} x_i y_j = a_{ij} y_i x_j$

Does equality hold when $x_i = y_i$ and $x_j=y_j,$ and $ i, j = 1, ..., n $.
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Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?