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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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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25 views

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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43 views

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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56 views

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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67 views

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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91 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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47 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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17 views

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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40 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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24 views

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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24 views

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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52 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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15 views

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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33 views

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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19 views

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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41 views

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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32 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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21 views

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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36 views

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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60 views

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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10 views

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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13 views

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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33 views

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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15 views

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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39 views

Relation between the frame bundle and the first jet bundle of a holomorphic vector bundle

Assume $\pi: E\rightarrow X$ is a holomorphic vector bundle of rank $r$ on an n-dim. complex manifold X and let $P\rightarrow X$ be the associated $GL_r(\mathbb{C})$-principal bundle of frames in $E$. ...
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34 views

why does the differential of this function on the manifold have this expression?

I am studying the proof of a theorem from Warner, where I am stuck in the following place : Let $\psi:M\longrightarrow N$ be an immersion and, let $m\in M$. Let $(W,\tau)$ be a coordinate system ...
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24 views

surjective differential at tangent vector is zero

The context is proving $deg(f,y)=0$ where $f:\partial M^{n}\to N^{n}$. page 2 at http://www.math.polytechnique.fr/~gravejat/SemiElev/Poincare-Hopf.pdf. Also,page 2 at ...
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45 views

Tangent plane to a surface

can you give an idea of how to beging this exercise? Let $S\subset\mathbb R^3$ a smooth surface, i.e., a 2-dimensional differentiable manifold. Show that for each $p\in S$, there exists an unique ...
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Covariant Derivatives of contravariant vector in curvilinear coordinates

$$D_mA^p = \partial_mA^p + \Gamma^p_{mn} A^n$$ so $$D_kD_mA^p = D_k(\partial_mA^p + \Gamma^p_{mn} A^n)$$ $$D_kD_mA^p = \partial_k(\partial_mA^p + \Gamma^p_{mn}A^n) + \Gamma^p_{kl}(\partial_mA^l + ...
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21 views

How to prove that the definition of saddle surface is affine invariant?

I have a smooth saddle surface in $\mathbb{R}^n$, so for any normal vector the second fundamental form of the surface has $\det \leq 0$. How can I proove that the surface is still saddle if I stretch ...
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48 views

laplace-beltrami and angular momentum operator

Im trying to understand this formula: In $\mathbb{R}^n$ we say that the laplacian can be seen as (radial part and angular part). : $$\triangle=(\frac{\partial^2}{\partial ...
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45 views

Involutive distribution on a submanifold

Let $M$ be a manifold and $D:M\to TM:x\mapsto T_xM$ a distribution on $M$. According to Frobenius theorem, $M$ admits a foliation by maximal integral submanifolds of $D$ if and only if $D$ is ...
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26 views

How to prove that a fiber bundle restricted to a nilmanifold has a one dimensional fiber?

If you have an orthonormal frame bundle $\pi:P \longrightarrow M$ and $S \subset P$ is a nilpotent manifold, how do you prove that $\pi:S \longrightarrow M$ is a subbundle with fibre of dimension ...
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Why does the expression of a point on a manifold seem to assume a coordinate system?

This question came up when I was studying the definition of natural coordinate functions. In many books, such as O'Neil, natural coordinate functions are defined as $u_i(p) = p_i$ where ...
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A question about complex polarization

Let $M$ be a smooth manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P \cap ...
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How to compute saddle point index using sourcing flow lines?

Prove $Index_{p}(\bigtriangledown f)$= "dimension of sourcing flow lines from p" ,where p is a critical point. Attempt Near $ p \in Cr(f) $ in some coord. $ f(x) - f(p) = $ $\sum_j x_j^2 - \sum_k ...
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Derive locally conformal neighborhoods between a Riemann surface and a diffeomorphic image of it

Consider a regular Riemann surface $M \subset \mathbb{R}^3$. A deep theorem in differential geometry states that any two regular surfaces are locally conformal [1]. The proof of this theorem consists ...
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72 views

Existence and uniqueness on this semi-linear parabolic PDE

I want to know whether the existence and uniqueness of a classical solution can be found about this question: Find a classical solution $u : [0,T]\times [0,\infty] \rightarrow {\mathbb R}$, such ...
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Central subgroups …

Have a question about the central subgroup. E is an elation group. Let $E_{x,y}$ be a stabilizer of points $x$ and $y$. Then the group $E_{x,y}$ is a central subgroup both $E_{x}$ and $E_{y}$, ...
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28 views

On implicit function theorem

We consider a manifold $M$ given by $f_{j}(x,y,v,w)=0$, $j=1,2$, where $f_{1,2}$ are smooth functions on $R^{4}$ verifying $rank \frac{D(f_{1},f_{2})}{D(x,y,v,w)}=2$ in every poin of $M$. If ...
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Connection between 'canonical projection' and 'implicit function solving' in implicit function theorems

Here are the two versions of the implicit function theorem (surjective/injective) commonly seen, but for these particular statements I took out from Differential Topology by Hirsch (P.214). ...
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22 views

Equivalent vector field with finitely many nondegenerate zeroes

This is an attempt to understand Milnor's proof that it's always possible to compute a vector field's index using a non-degenerate field. The strategy is simple: take $z$ to be an isolated zero of ...
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61 views

Polar coordinates and the chain rule

This is from the beginning of a proof of the isoperimetric inequality: ...we shall calculate $\ell(\gamma)$ and $A(\gamma)$ by using polar coordinates $x=r \cos{\theta}$, $y=r \sin{\theta}$. Using ...
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Banach manifold structure for sets of maps

I am looking at the lecture note of J.D.Moore, which is available at "www.math.ucsb.edu/~moore/globalanalysisshort.pdf". At page 16-17(just before Lemma 1.3.1), it explains the method to endow ...