Definition:Index Raising for Tensor

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Let $\struct {M, g}$ be a Riemannian manifold.

Let $\hat g : TM \to T^* M$ be the Riemannian metric mapping.

Let $F$ be a $\tuple {k, l}$-tensor.

Let $i \in \tuple {1, \ldots, k + l}$ be a covariant index position.

Index raising of $F$, denoted by $F^\sharp$, is an isomorphism $\sharp : \underbrace{V \times \ldots \times V}_{\text{$k$ times}} \times \underbrace{{T_p^*M} \times \ldots \times {T_p^*M}}_{\text{$l$ times}} \to \underbrace{T_p M \times \ldots \times T_p M}_{\text{$k + 1$ times}} \times \underbrace{{T_p^*M} \times \ldots \times {T_p^*M}}_{\text{$l - 1$ times}}$ such that:

$\map {F^\sharp} {\alpha_1, \ldots \alpha_{k + l}} := \map F {\alpha_1, \ldots, \alpha_{i - 1}, \alpha_i^\sharp, \alpha_{i + 1}, \ldots \alpha_{k + l}}$

where $F^\sharp$ is a $\tuple {k + 1, l - 1}$-tensor, and $\alpha_i$ is either a vector or a covector as appropriate.