# 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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### Locally Flat Understanding

I wanted to make sure that I was definitely understanding the proof of locally flat correctly. I can't see to find a similar proof to the one in the book, so I'm not super sure if my understanding/...
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### Doubt on Newman-Janis algorithm for a traversable Wormhole

Recently in a paper $$ the researchers presented a rotating traversable wormhole solution using the famous Newman-Janis Algorithm $$. But something is anoying me. In $$ they presented the ...
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### What are the local Lorentz transformations in general relativity?

What is the exact form of local Lorentz transformations (from the point of view of the metric) in a curved spacetime background like in general relativity? It should deviate substantially from ...
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### An identity between the d'Alembertian and the covariant derivative

Suppose $f$ is function which depends on $\phi$, $f = f(\phi)$; and $\phi$ is a scalar field. We define $$\square \equiv g^{\mu\nu} \nabla_\mu \nabla_\nu$$ and $$\nabla_\mu f(\phi) \equiv f_{;\nu}$$ ...
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### Covariant derivatives in a rank 2 tensor

I was trying to prove that for any second order tensor: $$A^{\mu\nu}_{;\mu\nu}=A^{\mu\nu}_{;\nu\mu}$$ considering the torsion free property and locally flat coordinates. Considering the point where ...
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### Conditions on one forms [closed]

I am trying to solve exercise 8.3 from Lightman's problem book, but I don't know where to start to get a sufficient and necessary condition on a field of one forms $\tilde\sigma$ for there to exist a ...
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### Is there any way to prove that contrarly to a flat 3D space, a curved 3D space can only be constructed in a 4D manifold?

This question is a result of me trying to understand how this universe can be possibly infinite if it isn't infinitely old. So to compare with an area that is flat it can be constructed both in 2D and ...
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### Conformal vector fields in $m$-dimensional Euclidean manifold

A vector field $X=X^\mu\partial_\mu\in\mathfrak{X}(M)$, where $M$ is a (pseudo-)riemannian manifold with a generic metric tensor $g_{\mu\nu}$, is a conformal Killing vector field if the conformal ...
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### What motivates defining vectors as first order differential operators?

I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
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### Does it make sense to take an infinitesimal volume of shape other than a cube?

The question clearer: Is the infinitesimal cube the absolute smallest infinitesimal volume? (Sorry if people thought that it meant: "Is it possible and is it done in daily life to use anything ...