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Curvature tensor of the induced connection on $Lambda^n_{mathbb{C}}(M)$
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I have read from a paper that if the curvature tensor of a connection on $TM$ is given locally by $sum alpha_jotimes A_j$, where $alpha_jin Gamma(Lambda^2(M))$ and $A_jin Gamma(mathrm{End}(TM))$, then the curvature tensor of the induced connection on $Lambda^n_{mathbb{C}}(M)$ is given by $-sumalpha_jotimes A_j^*$ where $A^*$ denotes the pull-back. Here $M$ is an almost-Hermitian manifold, and I don't understand why the curvature tensor of the induced connection has a minus sign. Could you please show me how to derive it?
differential-geometry complex-geometry
asked 22 hours ago
J. XU
429213
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up vote
0
down vote
favorite
I have read from a paper that if the curvature tensor of a connection on $TM$ is given locally by $sum alpha_jotimes A_j$, where $alpha_jin Gamma(Lambda^2(M))$ and $A_jin Gamma(mathrm{End}(TM))$, then the curvature tensor of the induced connection on $Lambda^n_{mathbb{C}}(M)$ is given by $-sumalpha_jotimes A_j^*$ where $A^*$ denotes the pull-back. Here $M$ is an almost-Hermitian manifold, and I don't understand why the curvature tensor of the induced connection has a minus sign. Could you please show me how to derive it?
differential-geometry complex-geometry
asked 22 hours ago
J. XU
429213
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have read from a paper that if the curvature tensor of a connection on $TM$ is given locally by $sum alpha_jotimes A_j$, where $alpha_jin Gamma(Lambda^2(M))$ and $A_jin Gamma(mathrm{End}(TM))$, then the curvature tensor of the induced connection on $Lambda^n_{mathbb{C}}(M)$ is given by $-sumalpha_jotimes A_j^*$ where $A^*$ denotes the pull-back. Here $M$ is an almost-Hermitian manifold, and I don't understand why the curvature tensor of the induced connection has a minus sign. Could you please show me how to derive it?
differential-geometry complex-geometry
asked 22 hours ago
J. XU
429213
I have read from a paper that if the curvature tensor of a connection on $TM$ is given locally by $sum alpha_jotimes A_j$, where $alpha_jin Gamma(Lambda^2(M))$ and $A_jin Gamma(mathrm{End}(TM))$, then the curvature tensor of the induced connection on $Lambda^n_{mathbb{C}}(M)$ is given by $-sumalpha_jotimes A_j^*$ where $A^*$ denotes the pull-back. Here $M$ is an almost-Hermitian manifold, and I don't understand why the curvature tensor of the induced connection has a minus sign. Could you please show me how to derive it?
differential-geometry complex-geometry
differential-geometry complex-geometry
asked 22 hours ago
J. XU
429213
asked 22 hours ago
J. XU
429213
asked 22 hours ago
J. XU
429213
asked 22 hours ago
J. XU
429213
asked 22 hours ago
J. XU
429213
429213
add a comment |
add a comment |
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