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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Generalization of Grassmann manifold to include translations?

I came across a certain generalization of Grassmann manifolds and was wondering what work if any has been done on it. If you take the space of $n\times p$ real matrices, $n>p$, and define an ...
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Integral over a Funnel in Fermi coordinates

Suppose we are in the Hyperbolic plane, defined as $$ \{w = u + iv \mid v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$ I am given a funnel $F$. This object is isometric to a ...
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Questions on a sub-bundle of $\mathbb R^3\setminus \{0\}$

Let us consider spherical coordinates $(r,\theta,\phi)$ on $\mathbb R^3$ and the manifold $M:=\mathbb R^3 \setminus \{0\}$. Let us consider the 1-form on $M$ $$ \omega = zdz ...
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A question on surfaces

If $S$ is the surface given by the function $z=y^2-x^2$, if I have the points $A=(1,0,-1)$, $B=(0,1,1)$, $C=(1,1,0)$, how can I use the Gaussian curvature to determine if there is an isometry of $S$ ...
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50 views

Laplace’s Equation in Hyperspherical Geometry

I've been reading this reference. I agree with everything they say but there's something that I can't really understand...They get that, for example n=2, the potential created by some source on a ...
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463 views

The Darboux vector is defined by $D = \tau T + \kappa B$. Show that $T' = D \times T$

The Darboux Vector is defined as $D = \tau T + \kappa B$. Show that for a unit speed curve $$T' = D \times T \hspace{1cm} ... $$ Here, the $...$ represents the fact that there are a few ...
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fibers are connected implies that total space is connected in a surjective submersion between manifold

Could anyone tell me how to prove the following problem?I have no idea! Thank you! If $f:M\to N$ is a surjective submersion in the category of smooth manifolds, if $N$ is connected, and if ...
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285 views

Describe tangent and normal bundle to a manifold

Consider the set $X:=\{(x,y,z)\in\mathbb{R}^3 | x^2+3y^2=1+z^2\}$ I have to show that $X$ is a submanifold of $\mathbb{R}^3$ (and this is trivial); then, using on $T\mathbb{R}^3$ the standard ...
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On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
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Can you extend vector fields on a manifold?

I know that not necessarily you can extend a smooth vector field defined over a subset of a manifold to ALL of the maniffold, but, can you extend it at least to an open set? (Of course I'm talking ...
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91 views

Confusion over notation in a book on the mathematics of QFT by Faria-Melo

While formulating this question, I arrived at a likely interpretation provided in an answer to my own question below. My problem appears to be one of inexperience in working with ambient coordinates, ...
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Approximation of Lipschitz functions on Riemannian manifolds

Let $ (M,g) $ be a Riemannian manifold ($ g$ Riemannian metric) and let $ f: M \rightarrow R $ be a Lipschitz function (with respect to $ g $) with compact support. I want to study if it is possible ...
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356 views

Vector field decomposition into gradient and hamiltonian vector field

I have just read (without further explanation) that any vector field $(v_x(x,y),v_y(x,y))$ from $\mathbb{R}^2$ to $\mathbb{R}^2$, which has a continuous derivative, can be uniquely written as the sum ...
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Why Local Minimum is calculated for a derivative function instead of actual function?

In Machine learning regression problem, why the local minimum is computed for a derivative function instead of the actual function? Example: http://en.wikipedia.org/wiki/Gradient_descent The ...
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354 views

Motivation behind the definition of Zariski tangent space

Intuitively I think of tangent space at a point as the set of all points lying in the tangent plane passing throug that point. Here is the definition of Zariski tangent space Let X be an ...
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224 views

Does projectivizing always fix problems at infinity? (Or, am I making a mistake somewhere?)

This question is motivated by the following homework problem. I'm trying to explicitly compute the homeomorphism $f:S^2 \rightarrow \mathbb{CP}^1$ by using stereographic projection and considering ...
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217 views

Concept and meaning of immersion

Who can explain concept and meaning of "Immersion" maps, very easy and useful? thanks for advanced.
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125 views

De Rham cohomology of $\mathbb R^3$ without lines and a circumference

I am trying to calculate De Rham cohomology of the following spaces: $X=\mathbb R^3\setminus r$ where $r$ is a line; $Y=\mathbb R^3\setminus (r \cup \gamma)$ where $r$ is a line and $\gamma$ is a ...
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Total Derivative and Multilinear Functions

So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review: Given a function ...
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51 views

Volume form on Hamiltonian level surface

Assume we have a Hamiltonian system on $(\mathbb{R}^{2n},\omega)$ with Hamiltonian $H = H(q,p)$. In a paper I read, it says, without clarification, that the natural Liouville measure $\mu$ obtained by ...
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79 views

Length of a curve on a manifold using diffeomorphisms

Lets say I have two (compact) manifolds $U$,$V$ and a diffeomorphism $\psi:U\rightarrow V $. The shortest way between two points $a$ , $b \in V$ is given by a parametrisation $\gamma :W \rightarrow ...
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Elementary Morse theory

I want to initiate myself to 'elementary' Morse theory and use it to calculate the Euler-Poincare characteristic of some compact manifolds (spheres and torus ...). I do not know what strategy should I ...
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Sufficient conditions for Hessian definiteness for critical points of functionals

Let $C$ be the set of smooth curves from the unit interval into $\mathbb{R}^n$. Let $f : C \rightarrow \mathbb{R}$ be a functional on these curves given by $f(x) := \int_0^1 L(x,\dot{x}) dt$. Define ...
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Restriction of a differential form to an isotropic submanifold

From Analysis and Algebra on Differentiable Manifolds, first edition, exercise 2.6.4., question 1 (slightly edited for this post): Let $\vartheta$ be the canonical 1-form on the cotangent bundle $T^* ...
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Prove $\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$

$$\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$$ The above is an identity frequently used in ...
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what are conormal distributions?

According to the first answer in this post, a conormal distribution $u$ on a manifold $X$ relative to a (closed, embedded) submanifold $Y$ is an element of a Banach (or Hilbert) space $H$ such that ...
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184 views

Understand the Hyperbolic space

I've been trying to find the expression for the metric of the hyperbolic n-space, $\mathbb H^n$. For $n=2$ I've found (e.g. here) that $$ds^2=\frac{dx^2+dy^2}{y^2}.$$ But for $n>2$ I can't seem to ...
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frontier of class $C^{1}$.

Studying the Divergence Theorem (Gauss theorem), found the definition of frontier of class $C^{1}$. Which means? That is, the one which is a set with boundary of class $C^{1}$? Can give reference ...
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105 views

Visualizing diffeomorphisms

This is probably a really basic question (hence my asking it here as opposed to MO). In a comment to a question on mathoverflow ...
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$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...
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Intuitive meaning of immersion and submersion

What is immersion and submersion at the intuitive level.what can be visually done in each case?
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Existence Theorem for Geodesics

The text I am reading now defined geodesics to be those curves that satisfy the following differential equation: $\ddot{\gamma}^k(t)+\dot{\gamma}^i(t)\dot{\gamma}^j(t)\Gamma^k_{ij}(\gamma(t)) = 0$ ...
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56 views

Show that $\kappa_{\delta} = \frac{\kappa}{1 - r\kappa}$

$\delta_r(t) = \gamma(t) + r U(t)$ is a parallel curve to a parametric curve $\gamma(t): I \rightarrow \mathbb{R}^2$ at distance $r$. I have already shown that $\delta_r' = (1 - r \kappa) \cdot | ...
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Riemannian metric. Help with notation.

I was just reading about the hyperbolic space (upper-half plane model) and i'm getting kind of confused about the notation for the Riemannian metric. The half-plane is defined as $$ H = \{(x,y) \in ...
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1answer
63 views

Nadirashvili surface (part 3)

The article that I'm considering is 'Notes sur la démonstration de N. Nadirashvili des conjectures de Hadamard et Calabi-Yau' by Pascal Collin and Harold Rosenberg. In the proof of the appendix (of ...
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Gradient of a Lipschitz function on a Riemannian manifold.

I'm referring to the article of D. Fischer-Colbrie and R. Schoen The structure of complete stable minimal surfaces in 3-Manifolds of non-negative scalar curvature (journal link, pdf). In the proof ...
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40 views

Nadirashvili surface (part 2)

The article is 'Hadamard and Calabi Yau conjectures on negatively curved an minimal surfaces' Nadirashvili. In the proof of proposition 4.3 it asserts that the function y is holomorphic. I'm not sure ...
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545 views

Showing that left invariant vector fields commute with right invariant vector fields

I'm trying to prove that if $G$ is a Lie group, $X$ is a left-invariant vector field on $G$, and $Y$ is a right-invariant vector field on $G$, then $[X,Y] =0$. When I imagine what it means to be ...
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The standard connection in a regular surface is symmetric

The concrete setting: Let $M\subset \mathbb{R}^3$ be a regular surface. A vector field $X$ in $M$ is a differentiable function $X: M\to \mathbb{R}^3$ such that $X(p)\in T_pM$. Here we are taking the ...
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170 views

elementary questions about differential forms

QUESTION 1: So I know that if $\omega$ is an alternating $p$-form for odd $p$ on some vector space $V$, then $\omega\wedge\omega = 0$. But...isn't the same true for any $p$? Ie, take for example $p ...
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What are all isometry classes of the 2-sphere?

In topology, one learns how to classify the compact surfaces up to homeomorphism. And in fact, since "homeomorphic" and "diffeomorphic" coincide in dimension 2, we can classify the compact (smooth) ...
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Are the Ricci and Scalar curvatures the only “interesting” tensors coming from the Riemannian curvature tensor?

In studying Riemannian geometry, one learns about the Riemann curvature tensor $$Rm(X,Y,Z,W) = \langle \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, W\rangle$$ and its various symmetries. ...
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Diffeomorphisms to $S^n$

Is $S^4$ diffeomorfhic to $S^2\times S^2$? Moreover. Is $S^n$ diffeomorphic to some cross product of manifolds $X\times Y$ for $n\geq2$? Is there a elemental topological invariant to let me see ...
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Nadirashvili surface

I'm referring to the article of N. Nadirashvili "Hadamard's and Calabi-Yau conjectures on negatively curved and minimal surfaces". In the proof of proposition 4.3 author use a theorem of Walsh. Now ...
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559 views

where can I find solutions to A comprehensive introduction to differential geometry by Spivak?

I have tried google and I fail to find solutions to the exercises in the book A comprehensive Introduction to differenial geometry volume I by Spivak. Does anyone know about a site with solutions to ...
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85 views

Why this equality must holds for minimal surfaces?

When minimizing a surface area with respect to a fixed volume $V$, I found in some notes that the parametrization $X: U \longrightarrow \mathbb{R}^3$ must satisfy the equality $\iint_U (2H - \lambda) ...
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Compute the differential of a smooth map

Let $S\subseteq \mathbb{R}^3$ be an oriented regular surface and let $N$ be a field of normal unitary vector on $S$. We consider the map $F:S\times \mathbb{R}\rightarrow \mathbb{R}^3$ defined by ...
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a doubt on manifold with boundary, critical point, space of jets etc

could any one explain me the following paragraph by a simple example? "a manifold with boundary is understood to be a smooth (real or complex) manifold with a fixed smooth hypersurface. Two functions ...
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1answer
230 views

Symplectic 2-Sphere

Consider the sphere $S^2\subset\mathbb{R}^3$ in cylindrical coordinates $(\theta, z)$ (away from poles $z=\pm 1$) with symplectic structure $\omega=d\theta\wedge dz$. I want to show that the vector ...
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how to cover a manifold almost everywhere?

I pose the following question: Given a connected (maybe we need compactness) manifold or regular surface, can we find a single parametrization $\chi:B\to M$ from the open ball to the manifold, such ...