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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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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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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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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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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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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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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Unbounded geodesics on a hyperboloid

What would the unbounded geodesics on a hyperboloid be? The meridians and circles are bounded, but not sure what the unbounded geodesics would be?
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Tangent bundle of n-surface in $\mathbb{R}^{n+1}$

i would like some help in this question bellow: Let $S$ be an oriented n-surface in $\mathbb{R}^{n+1}$ and let $T(S)=\{ v\in \mathbb{R}^{n+1}_{p} ; p\in S - and -v.N(p)=0\}$. Show that $T(S)$ is a ...
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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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31 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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23 views

vector bundles and cocycles

I need a detailed solution to a self-study book's exercise: "Show that two vector bundles on M are isomorphic iff their cocycles relative to some open cover are equivalent" I can show it in one ...
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18 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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11 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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29 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

If $\mathbb{X}, \mathbb{Y} \in \mathcal{X}(U)$, the set of smooth, complete vector fields on $U$ where $U$ is an open set. Let $\Phi_t,\Psi_s$ are their respective flows and let $\Gamma_{s,t}= ...
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78 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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21 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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40 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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$C^{\infty}$-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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Can you give a example about of curvature tensor

Can your give a Riemann manifold $(M^n,g)$,let $R(X,Y,Z,W)=g(R(Z,W)X,Y)$,and under some coframe $w^1,...w^n$, $$R=R_{ijkl}w^i \bigotimes w^j\bigotimes w^k \bigotimes w^l$$ such that,$\forall i,j ...
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20 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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37 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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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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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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21 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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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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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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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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Application of constant rank

Let $M^m$ and $N^n$ be differentiables manifolds, where $m$ is dimension of M and $n$ is dimension of $N$. If $f:M^n \to N^n$ is smooth map, with constant rank, show that: a)If $f$ is injective ...
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39 views

Pullback of Euclidean metric under spherical coordinates

I came across the following problem when I was reading a paper. Let $\Omega=B_2\subset\mathbb{R}^3$, which is the ball of radius 2 centered at origin. Map $F:B_2\backslash \{0\}\to B_2\backslash B_1$ ...
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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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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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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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whitney class of fiber bundles over classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $\rho: \Sigma_k\to GL(k)$ be regular representation by permuting basis. Let $\rho': B\Sigma_k\to BGL(k)=G_k(\mathbb{R}^\infty)$ be the induced ...
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Books or notes focusing more on the intersection between manifolds and topology?

I try to prepare for the qualify exams, and find that the problems of geometry part are quite interesting. In the past, I just learned some elementary things on manifolds and algebraic topology ...
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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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Produce one smooth curve on one triangle mesh

I hope to get one smooth curve on one triangle mesh. I get one path on the mesh at first. The path consists of vertices of the mesh. I can see the path from the image below. Each one green dot ...
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180 views

Why Shape space is manifold?

In Shape analysis, often shape is considered as continuous parametrized closed curve and the shape space as Hilbert Riemannian manifold. Can any one help me to understand, why the shape space is ...
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Inverse of the Stereographic projection $\mathbb{CP}^1 \to S^2$

I've some problems with this exercise: Consider the stereographic projection $$ \varphi \colon S^2 \setminus \{ (0,0,1)\} \to \mathbb{CP}^1\setminus \{[1:0]\}$$ $$ (x,y,z) \mapsto [ x+iy:1-z]$$ and ...
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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 $x$ ...