• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.371-376
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• 2008

Let $P(M,G,{\pi})=:P$ be a principal fibre bundle with structure Lie group G over a base manifold M. In this paper we get the following facts: 1. The tangent bundle TG of the structure Lie group G in $P(M,G,{\pi})=:P$ is a Lie group. 2. The Lie algebra ${\mathcal{g}}=T_eG$ is a normal subgroup of the Lie group TG. 3. $TP(TM,TG,{\pi}_*)=:TP$ is a principal fibre bundle with structure Lie group TG and projection ${\pi}_*$ over base manifold TM, where ${\pi}_*$ is the differential map of the projection ${\pi}$ of P onto M. 4. for a Lie group $H,\;TH=H{\circ}T_eH=T_eH{\circ}H=TH$ and $H{\cap}T_eH=\{e\}$, but H is not a normal subgroup of the group TH in general.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.793-798
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• 2013

Let ${\phi}$ be a map of a manifold M into another manifold N, L(N) the bundle of all linear frames over N, and ${\phi}^{-1}$(L(N)) the bundle over M which is induced from ${\phi}$ and L(N). Then, we construct a structure equation for the torsion form in ${\phi}^{-1}$(L(N)) which is induced from a torsion form in L(N).