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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

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Surface where number of coordinate charts in atlas has to be infinite

In the definition of a parametrised surface $S$, for every point in the surface, $p \in W \subseteq S$, where $W$ is open, there exists a coordinate chart or patch , $F :U\to \mathbb{R}^n$ that maps to $p$ from an open subset $U \in \mathbb{R}^n$ Is…
Samuel Tan
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Is the scalar curvature the only isometric invariant of a Riemannian 2-manifold?

Given two Riemannian Manifolds of dimension 2, and a point on each. If the scalar curvatures are isomorphic (as functions) in some neighbourhoods of these points, are then the manifolds necessarily locally isometric?
unknown
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How do I see that the tangent bundle of torus is trivial

I've been having a hard time trying to determine if the tangent bundle of a differentiable manifold is trivial. Namely, if there exists a diffeomorphism between the tangent bundle $TM$ of a given manifold $M$ and the product manifold of $M\times…
henryforever14
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Christoffel Symbols for Spherical Polar Coordinates

If we are given a line element; $$ds^2=dr^2+r^2d\theta^2+r^2sin^2\theta d\varphi^2$$ We can easily then see that the metric and the inverse metric…
arcturus7
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geodesic computation: "energy" minimization versus arc length minimization

Is it true that applying the Euler-Lagrange equation to the integral $E(\gamma)=\int_{t_1}^{t_2} g_{\alpha\beta}(\gamma^{\alpha})'(\gamma^{\beta})'\operatorname{d}\!t$ rather than the arc length integral $L(\gamma)=\int_{t_1}^{t_2}…
J. Heller
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Curvature $0$ and involutive horizontal distributions

I am trying to check why curvature $0$ implies that the horizontal distribution is involutive. Let $\pi:P\to U$ be a principal $G:=GL_n$ bundle. Assume that $P$ is trivial and $\pi$ admits a section. Thus, $P\cong U\times G$. A connection on $P$ is…
user5389
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What is an "allowable surface patch"?

From the book Elementary Differential Geometry, Andrew Pressley, Second Edition, the author defined an allowable surface patch is follows: If S is a surface, an allowable surface patch for S is a regular surface patch $\sigma:U \rightarrow R^3$…
Rafid
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Are homeomorphic differentiable manifolds actually diffeomorphic?

Let $M$ and $N$ be two n-dimensional smooth manifolds.Suppose their underlying topological spaces are homeomorphic through $f$. Does $f$ automatically become a diffeomorphism with respect to the given smooth structures? If not, can I adjust any of…
gp120
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Proving rigorously a map preserves orientation

In the following link: http://hilbertthm90.wordpress.com/2009/09/09/what-i-talk-about-when-i-talk-about-orientation/ They state that the antipodal map $f: \mathbb{S}^{n} \rightarrow \mathbb{S}^{n}$ is orientation preserving if n is odd. I'm trying…
user17182
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Why is the manifold structure on the tangent bundle unique?

...subject to the conditions that (i) the projection be smooth and that (ii) smooth sections correspond to smooth vector fields. This homework problem is really bugging the hell out of me. Of course it can be checked locally, so we'll look at an…
Aaron Mazel-Gee
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Some questions about differential forms

If $A$ is a differential one form then $A\wedge A .. (more\text{ }than\text{ }2\text{ }times) = 0$ Then how does the $A\wedge A \wedge A$ make sense in the Chern-Simon's form, $Tr(A\wedge dA + \frac{2}{3} A \wedge A \wedge A)$ ? I guess this…
Student
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Commutation of covariant derivative of functions

Le$f$ be a smooth function on a Riemannian manifold $M$. My questions are: a) If $\nabla_i f$ is a function, why is not true that $\nabla_j\nabla_k\nabla_if=\nabla_k\nabla_j\nabla_if$? This question arose when I wrote…
Myself
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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 about smooth extensions)
Sak
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What is the universal property of the tangent bundle of a smooth manifold?

The process of writing my own notes on smooth manifolds have led me to wonder about this. All I've really found is the following: In addition to Madame Ehresmann's references, there is in Spivak's Comprehensive Introduction... an abstract …
Eivind Dahl
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Is every compact hypersurface contained in a sphere which it touches twice?

Let $M\subset \mathbb{R}^{n+1}$ be a compact $n$-manifold. There exists, then, a smallest $n$-sphere containing $M$, and it must touch it in one point. Must it touch it twice? This seems quite intuitively right to me, but I've no idea how to prove…
Bruno Stonek
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