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- Homework Statement
- compute the metrics, Lie brackets, and covariant derivatives in the Levi - Civita connection of the 3-sphere

- Relevant Equations
- {x^2+y^2+u^2+v^2=1}

This week, I've been assigned a problem about a 3-sphere. I am confused how to approach this problem and any comments would be greatly appreciated.

(a) - would I be correct to assume the metric G is simply the dot product of two vector fields with dx^2 dy^2 du^2 and dv^2 next to their corresponding terms?

for example:

g(A,A) = (ydx)^2 + (xdy)^2 + (vdu)^2 + (udv)^2

g(A,B) = (uv)dx^2 + (xdy)^2 + vy(du)^2 + uv(dv^2)

why are the metric arranged in a 3x3 pattern?(b) computing the lie brackets with the chain rule

[A,B] = operate on B with A - operate on A with B

[B,C] = operate on C with B - operate on B with C

[A,C] = operate on C with A - operate on A with C

are these computations entirely independent of the metric g?

(c) does this part involve information from the Lie brackets?thank you in advance for ideas.

(a) - would I be correct to assume the metric G is simply the dot product of two vector fields with dx^2 dy^2 du^2 and dv^2 next to their corresponding terms?

for example:

g(A,A) = (ydx)^2 + (xdy)^2 + (vdu)^2 + (udv)^2

g(A,B) = (uv)dx^2 + (xdy)^2 + vy(du)^2 + uv(dv^2)

why are the metric arranged in a 3x3 pattern?(b) computing the lie brackets with the chain rule

[A,B] = operate on B with A - operate on A with B

[B,C] = operate on C with B - operate on B with C

[A,C] = operate on C with A - operate on A with C

are these computations entirely independent of the metric g?

(c) does this part involve information from the Lie brackets?thank you in advance for ideas.

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