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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derivative of the composition of two smooth map is the composition of derivatives

I can't type the LATEX so I uploaded the image. By Googling, I found that is the result of the pushforward, but there is no proof so I can't understand. ...
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How could I calculate displacement along a 2D polyline by integrating each dimension separately?

This question's field of application is GPS trajectory analysis, but I'll try to give it a more abstract mathematical treatment. Suppose the trajectory of an object in 2D plane is described by a ...
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Flow of Linear Vector Fields

The following is a statement from "Notes on the Topology of Vector Fields and Flows" by Daniel Asimov. In the case where a vector field on $\mathbb R^n$ is defined by a matrix, then there is a simple ...
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A question on self-adjointness of the shape operator

In the book Elementary Differential Geometry of Christian Bar, CUP, on the page 108, of the proof of theorem 3.5.5, the author wrote: Theorem: Let $S ⊂ \mathbb{R}^3$ be an orientable regular surface ...
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Understanding QEF's and assorted techniques for dual marching cubes

I'm attempting to implement dual marching cubes in a 3d engine and I'm getting lost in the math portion. http://www.cs.rice.edu/~jwarren/papers/dmc.pdf is the link to the original white paper. Part ...
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Local coordinates for a hypersurface

Let $M$ be a hypersurface in Euclidean space $E^{n+1}$, $e_1,e_2,..., e_n$ a frame field for its tangent bundle and $\theta_1,\theta_2,...,\theta_n$ are dual base. By knowing that $\theta_1$ is ...
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35 views

Theorema Egregium

I'm having trouble understanding what the Theorema Egreguim is. I read stuff on it but I'm not really grasping it. I know that it ties in with the Gaussian curvature but is there like a specific ...
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Meaning of fibered product

I need a small explanation about the next. If we write $p: TM\to M$ for the natural projection and $F$ for the natural bundle with $FM=p^{*}(T^{*}\otimes T^{*})M\to M$, then ...
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Foliation preserving diffeomorphisms for a codimension 1 foliation

I am studying reference frames on Minkowski spacetime $\mathcal{M}$, with (+,-,-,-) signature, from a differential geometric point of view, for this reason I came up with (codimension 1) foliations ...
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33 views

Can the second order frame bundle of a 2-d manifold be embeded inside the ordinary frame bundle of a 3-d manifold?

I would like to know whether the second order frame bundle of a 2-d manifold can be embedded inside the ordinary frame bundle of a 3-d manifold? Explanation Suppose we have a 3D smooth manifold ...
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27 views

References for Conjugate Points in Differential Geometry

I will have to give some lectures about conjugate points and I need some nice references about it, can anyone recommend me some? I already know manfredo's differential geometry of curves and surfaces ...
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40 views

Strictly convex boundary of Riemannian manifold

Let $(M,g)$ be a compact smooth Riemannian manifold with boundary $\partial M\subset M$. What does it mean to say that the boundary is convex and strictly convex? I can find definitions of ...
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spin^c structures and charged spinors

Given a spin structure and a complex line $\mathcal{L}$ we can form the tensor product of the complex spinor bundle $S$ and this line $S\otimes\mathcal{L}$. A spin^c structure attempts to construct ...
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131 views

Periodicity of Ideal Packings on Arbitrary Boundaries

This is a question that's been nagging me for some time that I find particularly interesting, but have no idea how to go about solving it. Consider some arbitrary solid $S$, such that $\partial S$ is ...
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18 views

Isotropy of transitive lie algebroid

Let $\rho:E{\longrightarrow} TM$ is a transitive lie Algebroid. I wanna show $Ker\rho$ is a vector sub-bundle of $E$ by introducing its bundle-chart. please hint me.
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28 views

dimension of tangent space to a boundary point of a convex shape

I have a basic question regarding the dimension of the tangent space at a point $P\neq0$ that lies on the boundary of a pointed convex cone with its point centered at 0. For a 3D cone that is ...
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26 views

express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
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Notation on an estimate of the sectional curvature.

In a paper on the Ricci flow i am currently reading (http://arxiv.org/abs/math/0612095) the following estimate occurs several times (for example Lemma 4.1 and 4.2); $$\operatorname{sec}(g_0) \geq ...
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How do i show that dy/dx does not equals to zero

The question goes like this. Equation of a curve is $2x^2-3xy+y^2=5$ Find the equations of the tangent and normal to the curve at point $(4,3)$. Show that there is no point on the curve at which the ...
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28 views

Why is the algebraic value of the covariant derivative equal to <dw/dt, N cross W>

Just a simple statement made by Do carmo. I suppose my linear algebra is rusty because I can't see how it's true. Although I understand why Dw/dt=[Dw/dt](N cross w).
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21 views

Characterization of reparametrizations of a straight line?

Let $\alpha$ be a regular curve. How to show $\alpha$ is a reparametrization of a straight line $t\mapsto p+tq$ if and only if $\alpha^{''}(t)$ and $\alpha^{'}$ are collinear?
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Global section $$s:T^*G\to Sp(T^*G)$$

Let $G$ be a Lie group and then how can we define the global section $$s:T^*G\to Sp(T^*G)$$ Where $Sp(T^*G)$ means symplectic frame bundle of $T^*G$.
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absolute value of torsion of asymptotic curve

Question:Prove the the absolute value of the torsion $\tau$ at a point of an asymptotic curve, whose curvature in nowhere zero, is given by $|\tau| = \sqrt{-K},$ where $K$ is the Gaussian curvature ...
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compact surface of revolution

I have to prove that a compact surface of revolution is diffeomorphic to a sphere or to a torus. And show that $\int_{S} K dA= $ =$\{4\pi,$ if S is spherical type 0, if S is toric type $\}$, ...
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How to prove a parallel $(u=u_0) $it self curvature?

My name is Gita, and I had aproblem with my math. and need help, I know that a parallel $C$ in a surface of revolution in $M$ be a geodesic if and only if $f'(u_0)=0$. and $C$ is non arc lenght ...
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How can we extend the proof that Lie groups are orientable to parallelizable manifolds?

Let $G$ be a Lie group with identity element $e$ and dimension $n$. I know we can take a non-vanishing $n$-form (call it $\omega$) on the vector space $T_eG$. For a Lie group, as we have an ...
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74 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
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45 views

How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
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integration in five dimensions space part three

I am following this: integration in five dimensions space part two Maybe I need to simplify my question: Find the integration of $\int_{\partial S}-p_1dq_1\wedge dp_2\wedge dq_2$, where $S$ is the ...
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39 views

Vertical vector fields on Principal G-bundle

I can see easily that every fundamental vector field on a principal G bundle is vertical, but can every vertical vector field be decomposed into a sum of fundamental vector fields? Thanks.
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29 views

How does this integration by parts work: $\int_{Q}v\varphi_t\;dxdt = -\int_S \varphi v|_{S} \nu_t - \int_Q v_t \varphi\;dxdt$

Let $\Omega(t)$ be a bounded domain for each $t$. Let $Q=\bigcup_{t \in [0,T]} \Omega(t) \times \{t\}$ and $S=\bigcup_{t \in [0,T]} \partial\Omega(t) \times \{t\}$. The normal vector to $S$ at ...
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finding tangent planes

Question: Determine the tangent planes of $x^2+y^2-z^2=1$ at the points $(x,y,0)$ and show that they are all parallel to the $z$ axis. Proof: Let the tangent plane of $f(x,y,z)=0$ at point $(x_{0}, ...
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Explicit diffeomorphism between two models of hyperbolic 3-space

I am looking for an explicit diffeomorphism between the matrix model and the homogeneous space model of hyperbolic 3-space. Since all models are isometric, this must exist. Here, I define the matrix ...
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50 views

Can Vanishing Cycles be Described as Fibers over Critical Points in a Lefschetz Fibrations?

I'm trying to see if it makes sense to see vanishing cycles in a Lefschetz fibration as the fibers over critical points. A Lefschetz fibration $f: M^4 \rightarrow X$ , where $M^4$ is a smooth ...
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46 views

Why the Oloid is developable surface?How to prove it?

It is well known that the Oloid is developable surface,but why?How to prove it?
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29 views

Components of a Vector Field

This may be quite a petty question, but it's been bothering me for a while. So for the vector fields $X$ and $Y$, we can write them in component form as $X=X^{\mu}\frac{\partial}{\partial ...
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Finding the Bessel function from its derivative

I have a situation: $A_k\frac{\partial J_m(k\rho)}{\partial \rho}=0$. where $k=k_1$ for $0\leq\rho\leq a$ and $k=k_2$ for $a \leq \rho \leq \Lambda-a$ with $a,\Lambda\leq \infty$. Can I proceed with ...
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Example of 1-dimensional hypersurface in $\mathbb{R}^2$ which is compact?

Is there an explicit example of a $1$-dimensional $C^k$ hypersurface in $\mathbb{R}^2$ which has no boundary and is compact? I know of a circle, but want something like an interval.
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When is a function of two variables positive?

There are two functions, $u(t)$ and $v(t)$, $u:[0,\infty]\to {R}$ , $t=\frac{1}{2}(y_{1}^{2}+y_{2}^{2})$, $v=\frac{-uu'}{2tu'-u}$. They should satisfy the next: $u>0$ and $u+2tv>0$. If we ...
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Tangent space of the tangent bundle 2

This is a continuation of the problem given in Tangent space of the tangent bundle which I repeat: Let $M$ be a differential $k-$manifold in $\mathbb R^n$, $g:TM\rightarrow\mathbb R$ given by ...
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31 views

Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
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22 views

Given vector F and a vector field A, calculate ∂AF

Let $F = x^4 − y^4$ and vector field $A = r \, \partial r$. Calculate $\partial AF$ in Cartesian coordinates. The answer I have been given is that $\partial AF = 4x^4 - 4y^4$. When I have tried this ...
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The tangent bundle over a manifold is locally trivial

Assume $M$ is an $m$-dimensional manifold and $(U,h)$ is a chart. Denote the differential of $h$ at the point $p$ by $T_ph$. How can I verify that $pr_1(Th(p,x))=h(p)$ where $pr_1$ is the projection ...
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Question concerning $e$-geodesic

I'm learning the book on Information geometry by Amari and Nagoaka after having taken a first course on differential geometry. My question is concerning a geodesic by the $\nabla^{(e)}$-connection. ...
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Tangent space of the tangent bundle

Let $M$ be a differential $k-$manifold in $\mathbb R^n$, $g:TM\rightarrow\mathbb R$ given by $g(x,v)=|v|^2$, where $|\cdot|$ is the usual norm in $\mathbb R^n$. Show that $Reg(g)=Reg(g_{\partial ...
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My mistake on proving “$deg(f,y)=0$ if f can be extented”.

Statement: Show $deg(f,y)=0$, when $f:\partial M^{n}\to N ^{n}$, y is a regular value, $\exists$ extension $F:M^{n+1}\to N$ and M and N are compact smooth mflds. The outline: 1) $F^{-1}(y)$ is a ...
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Orthogonal Array Planes

In an OA-plane (Orthogonal Array-Plane) how we define the touching of two lines? If we consider OA of strength of 3 with unit index , that is the plane OA-plane of rank 3 (we call this Laguerre ...
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Does the wave equation on manifolds evolve along geodesics?

For a compact, smooth surface $\mathcal{S} \subset \mathbb{R}^3$, consider the following wave equation: $$(\partial_{tt} + \Delta_{\mathcal{S}})u(t,x) = 0 \\ u(0,x) = \delta_y \\ u_t(0,x) = 0$$ ...
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Constructing a smoothly varying basis without singularities

I am trying to construct a smoothly varying and a differentiable basis to map a vector in $\mathbf{B}:\mathbb{R}^3 \to \mathbb{R}^3$. Given a vector field $\mathbf{n}(\mathbf{x})$ where $\mathbf{n} = ...
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Prove that the tangent space TpM is vector space .

We have to prove that $(T_p M , + , *)$ is vector space " + " : $T_p M \times T_p M \to T_p M$ $(X_p + Y_p)(f) = X_p(f)+Y_p(f)$ So we have to prove that $(T_p M ,+)$ is abelian group : ,, + " is ...