# Questions tagged [differential-geometry]

Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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### Gauss divergence theorem

I was reading new edition of Arfeken's book on Mathematical Physics I came to a two line state in the new edition in gauss divergence theorem section which was not included in the old edition and I ...
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### General Newton's second law [closed]

Thank you, I apologize: next time i will use LaTeX. I ask to Vincent Thacker: I read your post but I ask if it is possible to get your last equation (or similar) in classical mechanics starting by d2/...
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### Differential forms in projective space

I am currently reading some paper about the Amplituhedron, and it is using projective geometric way to present amplitudes. How can we define forms in projective space to measure volume for a polytope?
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### Definition of pull-back in gauge transformation

In an online lecture given by a famous physicist Youshi Wu, he says: Consider a map $g:\quad M\rightarrow G:x\rightarrow g(x)\in G$ ($G$ is a Lie-Group manifold, $M$ can be physical space or ...
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### Coefficient of an 1-form in position-representation of momentum operator where configuration space is NOT $\mathbb{R}^m$

I found this in the book Geometric Phase in Quantum Systems by A. Bohm et al. Where the position space representation of the momentum operator carries a (Where exactly my doubt is) coefficient of 1-...
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### What is the physical meaning of the exterior derivative of a non-holonomic constraint form?

Suppose we have a non-holonomic mechanical system, say Lagrangian, for example the Chaplygin sleigh is a model of a knife in the plane. Its configuation space is $Q = S^1\times \mathbb{R}^2$ with ...
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### Show that $\chi=\frac{1}{4\pi}\int _M d^2 \sigma g^{1/2} R +\frac{1}{2\pi}\int_{\partial M}ds k$ is locally a total derivative in 2D [closed]

In Polchinski Page 83 mentioned that $$\chi=\frac{1}{4\pi}\int _M d^2 \sigma g^{1/2} R +\frac{1}{2\pi}\int_{\partial M}ds k$$ is locally a total derivative in two dimension $R$ was the Ricci scalar ...
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### Looking for differential operators satisfying a specific commutation relation with the Laplace operator

Consider the Laplace operator on some manifold, $\Delta=(\det g)^{-1/2} \frac{\partial}{\partial x^j} \left( (\det g)^{1/2}g^{jk} \frac{\partial}{\partial x^k}\right)$. I am looking for differential ...
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### Global (not locally defined) quantities in General relativity?

So I was reading about center of mass (relativistic) on Wikipedia and saw an interesting comment: Since the Poincaré generators depend on all the components of the isolated system even when they are ...
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### Terminology: “Sum over Geometries” vs. “Sum over Topologies” in Quantum Gravity

I am currently studying loop quantum gravity and related approaches like simplicial quantum gravity, spin foam models, tensor models and group field theories. In texts about this topics, one often ...
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### Riemann tensor in Newman-Penrose formalism [closed]

In the NP (Newmann-Penrose) formalism, Riemann tensor is retarded time function $u=t-z$ $$R_{abcd}(u)$$ and Riemann tensor can take this form $$R_{abcd,p}(u)=0$$ I can't understand why this equation ...
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### Why in the curved space-time, the double derivatives of the position vector is symmetric but any other vector is not symmetric?

The double derivatives of the position vector (see image eq. (1)), connecting the two points in a curved space-time defined by the Schwarzschild metric, are symmetric under no torsion condition. This ...
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### Vectors as functions?

In my study of general relativity, I came across tensors. First, I realized that vectors (and covectors and tensors) are objects whose components transform in a certain way (such that the underlying ...
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### Poincaré bundle, a way of understanding it

So I will call the Poincaré bundle $(FM,\pi, M)$ the principal fiber bundle that has the Poincaré group as a structure group, the space of linear frames as total space and $M$ as the Riemann-Cartan ...
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### Change of coordinate vs change of reference axes

Does basis vectors change opposite to coordinate scaling ? For example, suppose I have some oblique coordinate system, and I decide to scale up both 'axes' by a factor of $a$ and $b$ respectively. The ...