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Tensor of type (k,l)

  1. Feb 9, 2010 #1


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    In John Lee's books "Introduction to smooth manifolds" and "Riemannian manifolds", a tensor of type [tex]\begin{pmatrix}k\\ l\end{pmatrix}[/tex] on a vector space V is defined as a member of

    [tex]\underbrace{V^*\otimes\cdots\otimes V^*}_{k}\otimes\underbrace{V\otimes\cdots\otimes V}_{l}[/tex]

    or as a multilinear function

    [tex]\underbrace{V^*\times\cdots\times V^*}_{l}\times\underbrace{V\times\cdots\times V}_{k}\rightarrow\mathbb R[/tex]

    or, when [itex]l>0[/itex], as a multilinear function

    [tex]\underbrace{V^*\times\cdots\times V^*}_{l-1}\times\underbrace{V\times\cdots\times V}_{k}\rightarrow V[/tex]

    (These three vector spaces are isomorphic). But in Wald's "General relativity", this is called a tensor of type [itex](l,k)[/itex]. I just want to ask, is this a "math vs. physics" thing, like when physicsts make their inner products antilinear in the first variable and mathematicians make theirs antilinear in the second? Or is there a standard convention that one of these guys is ignoring?
  2. jcsd
  3. Feb 18, 2010 #2
    i saw the type of (l,k) for both of math and physics.
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