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# 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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### Does General Relativity satisfy the Homotopy (or “h”) Principle?

By this I mean in the standard second order (whether metric or tetrad/verbein-based) form of General Relativity. I've been reading about the homotopy principle of late (see Eliashberg's introduction ...
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### Is there a physical interpretation of why Christoffel symbols do not transform like a tensor? [duplicate]

I understand mathematically why they don’t, but I was hoping someone could provide a physical interpretation to this. Is there a physical consequence of this fact?
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1 vote
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### Parallel transport in curved spacetime

This is sort of a very introductory question and I am not finding any reference regarding this. And let me know whether my answer is correct or not. For example, we are parallelly transporting a ...
1 vote
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### Generalized Stokes theorem in superspace

Do the generalized Stokes theorem apply in superspace? Any issues or uncommon behaviour of the gradient, divergence and rotational in superspace?
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### What does it mean to have a zero-dimensional induced metric?

I have an integral on the form \begin{equation} S=\int d^dx g_{\mu \nu} h^{ab}. \end{equation} In this example, $g_{ab}$ is a $d$-dimensional metric, $h_{ab}$ is an co-dim 2 induced metric. I wanted ...
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1 vote
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### Metric compatibility and torsion-free condition of GR

In an introduction to general relativity, we see the unique connection of a manifold is described by both the conditions, matric compatibility and torsion-free condition. The metric free condition can ...
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### Christoffel symbols and metric compatibility

In some coordinate system ,we can calculate the Christoffel symbols using the following procedure Basis vectors $\Gamma^k_{ij}\vec{e_k}=\frac{\partial \vec{e_i}}{\partial x^j}$ multiply $\vec{e_l}$ ...
1 vote
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### Path integral quantization for scalar, spinor, and Yang-Mills gauge fields on a general differentiable manifold?

Most of the recommendations on path integrals on differentiable manifolds I've found here (like Kleinert) focus only on formulating quantum mechanics via path integrals on differentiable manifolds. ...
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### Homogeneous Maxwell equations from the Bianchi identity

It is easily proven that: $$\partial_{\lambda} F_{\mu \nu} + \partial_{\mu} F_{\nu \lambda}+ \partial_{\nu} F_{\lambda \mu} = 0$$ Lots of sources say this equation implies the Homogeneous Maxwell ...
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### Christoffel Symbols for a Perturbed Metric

If a metric $g$ is given by the sum of a background metric $g_B$ and a perturbation $h$ ie. $g_{ij} = g_{Bij} + h_{ij}$, then the difference of the Christoffel symbols for the background metric and ...
1 vote
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### How is the full Riemann curvature tensor determined if the Einstein field equations only present the Ricci curvature tensor? [duplicate]

I'm currently learning general relativity following Leonard Susskind's lectures, and I was very surprised by the fact that the components of the full Riemann curvature tensor are relevant even though ...
1 vote
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### Closure of Lie brackets associated to Brown-Henneaux boundary conditions

When we impose Brown-Henneaux boundary conditions to the metric field on AdS$_3$, \begin{align} \begin{split} g_{tt}&=-\frac{r^2}{\ell^2}+\mathcal{O}(r^0)\,,\\ g_{t\phi}&=\mathcal{O}\...
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### What is a line always pointing at 45° on a sphere like?

I can easily imagine a line pointing dead vertically or horizontally on a sphere. Say I want to draw a line which is always pointing to some degree (eg 45°) from an origin. What is this line like? In ...
1 vote
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### How is this term an angle?

This question is regarding the $\Theta_{\mu\nu}$ term given in equation (3.8) of the paper (https://arxiv.org/abs/2206.07725). The term ( I have typed it below )is defined right below in (3.9) and ...
1 vote