dg.differential geometry – Calculate the 1 connection forms of a variety of deformed product using the method of moving frames

Let $ ( bar {M}, bar {g}) $ and $ ( dot {M}, dot {g}) $ to be two Riemannian varieties and let $ f in C ^ { infty} ( bar {M}) $ to be nowhere zero. Let $ (M ^ n, g) $ to be the curled product of both varieties with deformation function $ f $; C & # 39; is, $ M = bar {M} times dot {M} $ and
begin {equation}
g = bar {g} times_e point {g}: = bar { pi} ^ * bar {g} + (f circ bar { pi}) ^ 2 point { pi} ^ * dot {g}
end {equation}

or $ bar { pi}: bar {M} times dot {M} to bar {M} $ and $ dot { pi}: bar {M} times dot {M} to dot {M} $ are the natural projections. (If you like, we can simply write $ g = bar {g} + f ^ 2 dot {g} $).

Convention on Indices: $ 1 leq a, b, c, cdots leq q $ for $ ( bar {M}, bar {g}) $, $ q + 1 leq alpha, beta, gamma, cdots leq n $ for $ ( dot {M}, dot {g}) $ and $ 1 leq i, j, k, cdots leq n $ for $ (M, g) $. The summation convention of Einstein is supposed.

Let $ left { bar { omega} ^ a right } _ {a = 1} ^ q $ and $ left { dot { omega} ^ { alpha} right } _ { alpha = q + 1} ^ n $ to be local orthonormal coframes on $ ( bar {M}, bar {g}) $ and $ ( dot {M}, dot {g}) $ respectively. Then a local orthonormal coframe on $ (M, g) $ can be given by
begin {align}
omega ^ i: = left {
begin {array} {ccl}
bar { pi} ^ * bar { omega} ^ i & mbox {if} & 1 leq i leq q \
(f circ bar { pi}) dot { pi} ^ * dot { omega} ^ i & mbox {if} & q + 1 leq i leq n
end {array}
right.
end {align}

(Again, if you want, we can just write $ bar { omega} ^ i $ and $ f dot { omega} ^ i $ respectively). Moreover, under these two coframes, let's designate the connection forms-1 by $ bar { omega} ^ b_a $ and $ dot { omega} ^ { beta} _ { alpha} $ respectively.

My goal is to calculate the connection 1-forms of $ (M, g) $. By taking an outside derivative of the definition of $ omega ^ i $, applying Cartan's first structural equation and gathering the terms in the LHS, I arrive at
begin {gather}
omega ^ b wedge big ( omega ^ a_b- bar { pi} ^ * bar { omega} ^ a_b big) + omega ^ { beta} wedge omega ^ a _ { beta} = 0
\
omega ^ b wedge left ( omega ^ { alpha} _b- frac {(f circ bar { pi}) _ b} {f circ bar { pi}} omega ^ { alpha} right) + omega ^ { beta} wedge big ( omega ^ { alpha} _ { beta} – dot { pi} ^ * dot { omega} ^ { alpha } _ { beta} big) = 0
end {gather}

or $ (f circ bar { pi}) _ i $ is defined via $ d (f circ bar { pi}) = (f circ bar { pi}) _ i omega ^ i $.

It is now tempting to conclude directly from above that
begin {align}
omega ^ a_b & = bar { pi} ^ * bar { omega} ^ a_b
\
omega ^ { alpha} _ { beta} & = dot { pi} ^ * dot { omega} ^ { alpha} _ { beta}
\
omega ^ { alpha} _b & = frac {(f circ bar { pi}) _ b} {f circ bar { pi}} omega ^ { alpha}
end {align}

but I do not think this is generally valid for the sum of corner products. Cartan's lemma tells us that at most, we can only conclude that the expressions in parentheses can be written as a linear combination of $ omega ^ i $& # 39; s (it's trivial here) with coefficients satisfying some symmetry on the indices.

Therefore, I would like to ask how to proceed. How to continue to calculate the 1-forms connection? All comments, tips and answers are welcome.