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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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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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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35 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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72 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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91 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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64 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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61 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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41 views

Diagonalising Laplace--Beltrami on a Lorentzian Manifold

Is the Laplace--Beltrami operator on a Lorentzian manifold always diagonalisable?
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45 views

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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49 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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14 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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41 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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58 views

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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147 views

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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47 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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41 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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58 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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30 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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20 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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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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33 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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41 views

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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62 views

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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30 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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61 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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32 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 ...
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32 views

Help with terminology

I need some help unraveling the terms that appear in the following passage. I found it in a book on some conference proceedings related to Differential Geometry. Let $f:X \to R^3$ be a smooth curve ...
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63 views

Why is the space of symmetric positive definite matrices under the affine invariante metric a symmetric space?

I consider here the space of symmetric positive definite matrices SPD(n) with the metric invariant under: $(G,M) \rightarrow GMG^t $, where $M\in SPD(n)$ and $G\in Gl(n)$. The isotrpie group of $I$ is ...
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22 views

Verifying hypothesis (definition of Normal vector of a curve)

Let $f:I\subseteq \mathbb R\to \mathbb R^n$ a vector valued function. When we define the normal unit vector as: $N=T´(t)/||T´(t)||$, $T´(t)\neq 0$ $\forall t\in I$ ($T(t)$ is the unit tangent vector ...
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30 views

Rectifying linearly independent vector fields

Suppose we are given two vector fi elds $V_1$ and $V_2$ - defined on $R^n$- such that the vectors $V_1(x)$ and $V_2(x)$ are linearly independent for each $x$. Is it possible to find a diff ...
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question on tangent bundle

Let $X$ be a manifold and consider its tangent bundle $T(X)$ and let $p$ be the usual map $T(X) \to X$. Then why is it locally trivial ? i.e why for all $x\in X$ exist open neighborhood $U$ of $x$ ...
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34 views

Unit circle circumferential length on a Beltrami pseudosphere

What is the perimeter length of a geodesic circle of unit radius of tangential curvature? .. assuming that the circle lies entirely on one side of its cuspidal equator.
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Guessing shape made by Beltrami

If $ z = r^2 f(n \theta ) $ has constant negative Gauss Curvature, find $ f(\theta) $ or an ODE leading to it, when n is an even integer. I lost my earlier derivation involving elliptic integrals. ...
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Frobenius theorem for singular foliations , what are hypotheses impose over an endomorphism $P:TM\to TM$ that it span a singular foliation in M?

Let $M$ a smooth manifold. Given a morphism $P:TM\to TM$, i.e, $\pi\circ P\equiv Id$ and is linear over the fibers. If we suppose that $P^2=P$, then for each $x\in M$, $P_{x}:T_xM\to T_xM$ is a ...
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28 views

General Tensor Assistance

Sorry if this is a stupid question, but it might help me grok things if I can connect from something that's intuitive to me. Consider a transformation from Cartesian coordinates to spherical ones: ...
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45 views

Geometric Interpretation of QFT Scattering Integrals

Let $f\in C^\infty(\mathbb{R}^n,\mathbb{R}^k)$, and $g\in C^\infty(\mathbb{R}^n,\mathbb{R})$, where $k<n$. How do I compute $$\int_{\mathbb{R}^n}\delta^k(f(\mathbf{x}))\cdot ...
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Standard complex structure in contact geometry

In symplectic geometry there is the standard model: $(\mathbb{R}^{2n}, d(\sum_{i=1}^{n}x_{i}dy_{i}))$ with standard almost complex structure $ J_{0}= \begin{pmatrix} 0 & -E_{n} \\ E_{n} & 0 ...
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The second cohomology of total space of the $\mathbb CP^1$ bundle

$X$ is a closed smooth surface with $L$ a complex line bundle on $X$. Consider the $\mathbb CP^1$-bundle $P(L\oplus 1)$, that is the projectivization of the sum of $L$ and the trivial line bundle on ...
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Rerformulation of a previous question concerning a problem in physics that involves integration of 2-forms over the sphere

In this question the integral proposed in the posting concerns a physical problem that can shortly be described by the following : Let $J$ be a real valued function on the sphere (in fact it is a ...
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71 views

geodesic polar coordinate parallel circles

When is it possible to have the same constant geodesic curvature on all parallels of a constant Gauss curvature surface? EDIT: picture added.
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Visualization of Gauss Bonnet geometric objects

Where can we get to see some individual surface/line combinations in isometry visualizations with constant $ \int k_g ds $ (say total tangential rotation) ? Or with constant integral curvature ...
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21 views

How To Find a Set of Points Farthest Apart Within 3D Solid

I am trying to find out a method to solve the following problem: There are two parameters: 1) There is a solid 3D region plotted in a cartesian coordinate system. 2) There is a number of points that ...
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35 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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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 ...