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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Exactness of a certain sequence

Let $M$ be a connected manifold of dimension $m$ and let $\beta\in H^{1}(S^{1})$ such that $\int_{S^{1}}\beta = 1$. Let $\pi_{1}:M\times S^{1}\rightarrow M$ and $\pi_{2}: M\times S^{1}\rightarrow ...
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25 views

Euler Lagrange of a Curve

Let $C(s) = (x (s), y(s))$ be a closed curve inside a plane where $s$ is the parametric arc length parameter. What is Euler Lagrange equation for the following functional $$-\int_0^L \nabla C ds$$ ...
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28 views

Weierstrass-Enneper representation formula

State and prove the Weierstrass-Enneper representation formula. I have tried to find in some books but failed. I will be thankful if some one help me out of this.
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Change of variables and relations between partial derivatives.

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a map given by $f(s, t) = (s, st)$. Do we have the following identity \begin{align} \left( \begin{matrix} \frac{\partial}{\partial s} \\ ...
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32 views

Natural curvature tightening of parametric curve

I'm looking to compute the "tightening" of curvature for a curve (mine is mainly 2D but could be of any dimension). In particular, since I am mainly 2D, I'm staying away from the cross-product based ...
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28 views

Prove that the torus is a regular subvariety of $\mathbb{R}^3$

Consider the inclusion of the torus in $\mathbb{R}^3$, prove that the torus is a regular subvariety. I have the next idea. If we describe de points of the torus as pairs $(u,v)$. We have to prove ...
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fibration sequence of projective spaces

Question~1: How to construct a fibration sequence $$ S^3\to S^2 \to \mathbb{C}P^\infty\to \mathbb{H}P^\infty ? $$ Does $$S^3\simeq \Omega \mathbb{H}P^\infty ? $$ (Since $\mathbb{C}P^\infty\simeq ...
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32 views

Gaussian curvature proof

I can show the first part but not sure how to proceed after that.
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16 views

Surjective $\gamma \colon I \to M^1$, $\gamma (t_1)=\gamma (t_2)$ can be extended to a periodic parametrization of $M^1$

Suppose that $\gamma \colon I \to M^1$ is a smooth surjective curve in a Riemannian connected 1-dimensional manifold. Furthermore, suppose that it is parametrized via arc lenght i.e.:$$||\dot ...
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20 views

Symmetry of Christoffel symbols of the second kind

I was reading the article: http://physicspages.com/2013/12/22/christoffel-symbols-symmetry/, and I do not understand this: In the locally flat frame, this equation reduces to $\displaystyle ...
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42 views

Stuck in Preissmann's theorem

I am stuck on following the proof of Preissmann's theorem, whose statement is that Let $(M,g)$ be a closed connected Riemannian manifold of negative sectional curvature. Then every nontrivial ...
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55 views

Application Question - American universities strong in Differential Geometry?

Can anyone recommend some American universities (except those top 10 ones such as Harvard, Princeton, SUNY and Umichgan etc. ) which have departments with a solid focus on Geometry and Topology, ...
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Poisson bivector on the product of two manifolds

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by \begin{align} \pi_X = \sum_{i,j} ...
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If the linearization of a section is surjective on a slice, is the image under a submersion also a smooth manifold?

Let $V \rightarrow M \times N_1 $ be a vector bundle, where $M$ and $N_1$ are smooth manifolds and $s: M \times N_1 \rightarrow V$ a smooth section such that if $s(p,q) =0$ then $$ \nabla ...
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How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
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17 views

alternate definition of winding number?

If c is a singular $1$-cube in $R^2-\{0\}$ with $c(0)=c(1)$ , show that there is an integer $n$ such that $c-c_{1,n}=\partial c^2$ for some $2$-chain $c^2$. Here $c_{R,n}=(R\cos 2\pi nt,R \sin ...
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25 views

Useful Coordinate Families on Lie Groups

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. We all know, since $\exp$ is a diffeomorphism in some neighborhood $V$ of $0\in\mathfrak{g}$, that we can cover $G$ in coordinate charts ...
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Compact hypersurface in $\mathbb{R}^n$

Let $S$ be an $(n-1)$ dimensional hypersurface in $\mathbb{R}^n$. If we say that $S$ is compact, does this necessarily mean that $S$ has no boundary? Eg. $S$ can be a sphere but not a sphere cut in ...
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65 views

Generalized Stokes theorem applied to Tensor Moments

I am working with geometry and need to calculate 0th, 1st and 2nd moments in polyhedra, its polygons and its lines. From a previous answer in this forum, I understand that p-moments are: $ M^p = ...
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55 views

Total curvature of a piecewise smooth planar curve

I am trying to show that the total curvature of a piecewise smooth planar curve $\alpha$ has the property that $\displaystyle \int_{C}\kappa $ $\text{d}s + \displaystyle \sum_{j=1}^{l} \epsilon_{j} ...
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41 views

Hamiltonian Dynamics and the canonical symplectic form

1- What kind of advantages does one have by having a canonical symplectic form on $T^*M$ apart from the form being exact? Would it for instance provide any advantage to studying Hamiltonian dynamics ...
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46 views

Distance function under diffeomorphism of manifolds

community, I am concerned with measuring distances of systems under diffeomorphisms. Concrety, I consider a smooth diffeomorphism $\varphi: M\rightarrow N$ from the smooth differentiable manifold ...
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36 views

Diagonalising Laplace--Beltrami on a Lorentzian Manifold

Is the Laplace--Beltrami operator on a Lorentzian manifold always diagonalisable?
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Spaces of constant curvature

Can someone please provide a reference for the theorem that states that, up to isometry, there are only three isotropic spaces of constant curvature, E^n, S^n and H^n, in any dimension.
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Locally isometric if first fundamental forms agree

I am reading my notes on Geometry, and find this lemma: Two surfaces are locally isometric if we can find regular parametrizations with the same FFF. My question is does it really require the ...
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38 views

Distance on a 3-sphere

The arc-length $l$ between two points on on a 2-sphere of radius $R$ is given by $l=R\theta $ where $\theta$ is the subtended angle. I can rewrite this in terms of the euclidean distance $d$ between ...
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22 views

Restrictions for Green's Theorem?

a) Why does C have to be simple? I mean the difference in circulation should be negligible if the curve only crosses itself once right? Shouldn't the condition be the curve can only cross itself a ...
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12 views

Gauge theory H(P) and V(P)

In general you get a connection out of a one-form $\omega$ where $\omega = g^{-1}dg+g^{-1}Ag$ and $A= A^{\alpha}_{\mu}\frac{\lambda_{\alpha}}{2i}dx^{\mu}$ is given and the base ...
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35 views

Co-ordinate chart and components of a vector field.

Q) Using a coordinate chart, give a formula for the components of the vector field $[v,w]$ in terms of the components of $v$ and $w$. Where $[v,w]: f \mapsto v(wf) - w(vf)$ I don't know what the ...
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Non-commuting flows and obtaining a new expression about the pullback of a function

Let $U \subset \mathbb{R}^n$ and be an open set. If $\mathbb{X}, \mathbb{Y} \in \mathcal{X}(U)$, the set of smooth, complete vector fields on $U$. Let $\Phi_t,\Psi_s$ are their respective flows and ...
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Lie groups, Lie algebra and left invariant vector fields

Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let $a$ and $g$ be elements of a Lie group $G$, the left ...
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23 views

Position Vector - Using Geometry

Given Data and Specifications in the problem $O_i,O_j$ represent two co-ordinate systems with base orthogonal frame vectors $x_j,y_j,z_j$ and another frame $x_i,y_i,z_i$ with respect to that ...
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41 views

Poincare disk and mobius transformation

I have following problem Consider Poincare disk. $i.e$ $\mathbb{M}=\{ (x,y) \in \mathbb{R}^2 : x^2+y^2 <1 \}$ with metric $ g= 4\frac{dx^2 +dy^2}{(1-x^2-y^2)^2}$ Show the complex mobius ...
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38 views

$C^{\infty}$-homotopy type of the Moebius band

The Moebius band $N$ has the same $C^{\infty}$-homotopy type of $S^1 \times \mathbb{R}$. What is the explicit expression of the $2$ $C^{\infty}$-homotopies involved ?
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21 views

Christophel Symbols and planar

How can we get torsion from the christophel symbols? I want to show something is planar and am using christophel symbols, but how can I get torsion?
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44 views

Derivations on a Manifold

Let $M\subset \mathbb{R}^n$ be an $m$-dimensional manifold (in the ordinary Euclidean sense). Given a point $p\in M$ we define a derivation $D_p$ (at $p$) to be linear functional on the space of ...
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23 views

On definition of alternating multilinear form

I am still trying to understand differential forms. I understand that locally a differential $p$-form is a alternating multilinear form $T_xM \times \dots \times T_xM \to \mathbb R$ where the domain ...
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19 views

Cutting a surface at critical levels produces cylinders

Let $F$ a closed surface with isolated critical points and a homeomorphism $g: F \rightarrow F$ that maps critical levels to critical levels. Let us cut the surface $F$ by the critical levels. Do we ...
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operation on non-isometric surfaces with same Gauss curvature

SameK_diffEFG Apart from above text-book example are there more $ ds^2 $ metrically paired examples? I.e., surfaces with same K(u,v) as functions of both u and v? Without agreement of first ...
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About differential function between the sphere without poles and the hyperboliod of one sheet.

Let $S^2$ the unit sphere with the origin as center and $H=\{(x,y,z)\in \mathbb{R}^3:x^2+y^2-z^2=1 \}$. Denote by $N$ and $S$ the north and south pole respectively, and let ...
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Definition of the first Chern class in terms of the Ricci form

From B, B & S - String Theory and M-Theory: What does the square bracket mean? Obviously since $\mathcal{R}$ is a form and $c_1$ is a number, $[.]$ has to be an operator on forms ...
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24 views

Negative sign in the torsion of a curve

Why does the torsion of a curve has a negative sign in the formula $$\tau = -N\cdot (B)´$$ ? where N is the normal vector and B the binormal vector. My teacher didn´t explain it. I would appreciate ...
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A regular surface.

A half-line $[0,\infty)$ is perpendicular to a line $E$ and rotates about $E$ from a given initial position while its origin $0$ moves along $E$. The movement is such that when $[0,\infty)$ has ...
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Loxodrome : found an error on wolfram MathWorld web site?

Could it be?... We find this claim on Wolfram MathWorld site http://mathworld.wolfram.com/SphericalSpiral.html The claim is that this curve (given in oblate spheroidal coordinates in the limit where ...
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Proving curve devides a sphere into two equal-areas

let $\gamma$ be a closed geodesic without points of self-intersection on a closed convex surface. Prove that the spherical image of $\gamma$ divides a sphere into two parts with equal areas I ...
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18 views

Loxodrome equation

On Mathworld web site, the loxodrome is defined in oblate spheroid coordinates (even though with a peculiar "minus" sign in z, see here http://mathworld.wolfram.com/SphericalSpiral.html) where they ...
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how to test whether cobordism exist between two manifold or two system of polynomials

from wiki Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. from book geometrisation of 3-manifolds ...
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52 views

Finding global sections of a vector bundle over a compact manifold which generate each fiber

The following is an exercise I am doing for review for a midterm exam in differential geometry. Question: Let $\pi:V\to X$ be any real differentiable vector bundle of rank $r$. Assume that $X$ is ...
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23 views

Bianchi identity number of independent equations

What is the number of independent equations of the second Bianchi identity: $$R_{abcd;e}+R_{abec;d}+R_{abde;c}=0$$
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Rotating Frames on Curve

We are describing a curve with a moving frame. Figure 1 says about the definition of curve Figure 1 Specifications and Proprieties of the curve We can make a rotation 3D Matrix $R(s)_{3 ...