Hi, given a connection on the tangent space of a manifold, one can define its torsion: $$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$ What is the geometric picture behind this definition—what does torsion measure intuitively?
![intuition - What is torsion in differential geometry intuitively? - MathOverflow](https://cdn-ak-scissors.b.st-hatena.com/image/square/10df5a86e3137a3cd7bea6fd8ae7e36216ff6c05/height=288;version=1;width=512/https%3A%2F%2Fcdn.sstatic.net%2FSites%2Fmathoverflow%2FImg%2Fapple-touch-icon%402.png%3Fv%3Df1c9606b77ff)
I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows- An engineer, a physicist, and a mathematician are discussing how to visualise four dimensions: Engineer: I never really get it Physicist: Oh it's really easy, just imagine three dimensional space over a time- that adds your fourth dimension. Mathematician: No, it's way easier than that; just
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