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.
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How to produce symplectic manifolds which arise as regular level sets?
What kinds of equations $f(x) = 0$ give rise to symplectic manifolds? From what Gromov has stated in a few places, the most powerful way to construct manifolds is by looking at solutions of equations $...
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Does a self intersecting curve contain a simple curve
I got to this problem as part of a different problem I'm working on, not from any homework set. Any solution or reference from any field might help.
Say I have a smooth closed curve $\gamma:[0,1]\to\...
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When is the Simple Tensor Product of Non-Zero Alternating Multilinear Forms Alternating?
I am exploring the relationship between the tensor product ($\otimes$) and the exterior product ($\wedge$) of multilinear alternating forms (differential forms/covectors).
Let $V$ be a vector space ...
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Wedge product in characteristic r [closed]
I very much prefer in characteristic zero the Kobayashi-Numizu definition of wedge product in terms of the Alternator Alt without the coefficient (p+q)!/(p!q!), but reading Warner I realized that the ...
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Induced Orientation of $\partial M$ in Munkres’ "Analysis on Manifolds"
To ensure that a constructed collection of coordinate patches qualifies as an orientation, we must verify that it is maximal. However, Munkres claims “one can obtain an orientation of $\partial M$ by ...
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Looking for Sources that Reconcile Differential Geometry with Calculus of Variations
My question might be eyewash. If not, then I am unsurprised if this topic is low-hanging in the world of analysis and geometry; I've yet to see it with my greenhorn eyes.
My reading background:
I've ...
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The reason of definition of differentiability of maps between regular surfaces
In do Carmo's "Differential Geometry" there is definition of a differentiable function on a regular surface:
"let $f:V\subset S\to R$ be a function defined in an open subset $V$ of a ...
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Necessary condition for derivative of inverse function to exist? - inverse of inverse function theorem
I'm currently studying differential geometry and has encounter a probably absurd question regarding derivate of a smooth function, basically looking for inverse statement of inverse function theorem.
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Question on a proof by Spivak
I am studying the proof of the following theorem from Spivak's book “A comprehensive introduction to differential geometry"
In the last passage of the proof (I think), Spivak uses the change of ...
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Coordinate-free proof of uniqueness of tautological 1-form
Let $M$ be a differentiable manifold and $\pi\colon T^*M \to M$ be it's tangent bundle. The tautological 1-form $\theta$ on $T^*M$ is defined as follows: given a point $(x,p) \in T^*M$ (that is, $x \...
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Non orientability of a manifold
I am studying differential geometry. I know that an $m-$ differentiable manifold $M$ is said to be orientable if and only if it admits an oriented atlas $\mathcal{A}$, that is: for all $(A,\varphi)$, $...
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How far can an infinite number of unit length planks bridge an obtuse-angled gap?
This is inspired by the recent problem about bridging a right-angled gap with unit length planks.
What if the angle is $\frac \pi2+\epsilon$ instead of a right angle?
Intuitively, I'd expect the ...
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Confusion regarding Tangent Basis
I am trying to get a better grasp of how to find the basis of the tangent space. Here is one example I worked on in hopes of practicing it:
Consider the chart $(U,\psi)$, the manifold $\mathcal{M} = S^...
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Determining the location of portals using air pressure
Let me know if this is more on-topic for physics.se (or more generally, off-topic for mathematics.se).
Okay, imagine you have a volume of space with instruments measuring air pressure (and for ...
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Connected components on a regular manifold
Let $M = \{(x_{1}, x_{2}, x_{3}, x_{4}) \in \mathbb{R}^{4}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2} = -1\}$. Prove that $M$ is a regular submanifold of $\mathbb{R}^{4}$ with dimension 3. Compute the ...
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