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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Geometrical interpretation of curvature invariants

Consider a Riemannian manifold. It is possible to describe it by curvature invariants. Now, is there any geometrical description (intuition) for simple invariants such as scalar curvature, Ricci ...
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Can a “time dimension” be part of a spherical topology?

I've heard it speculated that the spatial dimensions of the universe is a 3-sphere. Or a 3-torus. But usually, I guess, it's assumed that the "time" dimension just has its own geometry, like a line, ...
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Matrix Expression of the Maurer-Cartan Form

I'm looking for clarification re: the 'classical' matrix expression for the Maurer-Cartan form $$g^{-1} dg$$ (I have seen the related posts, they don't answer my specific question.) Specifically I ...
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Lagrangian form of acceleration

Reading the Wikipedia article on Lagrangian mechanics I have problems to understand one basic proposition in the motivation of Euler-Lagrange equation. The article says: For me is unclear how the ...
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A Potential Euler-Lagrange Equation Alternate Derivation?

Can the Euler Lagrange Equation be derived with this overall strategy? Step 1 – Define a geodesic in flat space to be $\frac{d}{dy} \frac{ds}{dx} = \frac{d}{dx} \frac{ds}{dy}$, where $ds$ represents ...
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