I wonder why first making person who express $$\lambda^\mu_{\;\; \nu}$$ made like that? although $$\lambda^{\mu\nu},\lambda_{\mu\nu}, \lambda^{\;\;\mu}_\nu$$ are also possible!! please teach me obvious reason!!
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I have never seen these symbols, with one subscript and one superscript, typed with so much horizontal separation between the two. As far as history, some aspects of this are referred to as the Einstein convention. However, calculations in differential geometry did occur before Einstein, for example Christoffel, Ricci, Levi-Civita. – Will Jagy Jan 30 '18 at 03:52
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1It helps to remove ambiguous answers when we raise and lower indices, because in general $\lambda_{\mu \nu} \neq \lambda_{\nu \mu}$. When you hit it with a metric tensor to lower an index $\lambda^{\mu}{\nu}$ can either be $\lambda{\mu \nu}$ or $\lambda_{\nu \mu}$ – Triatticus Jan 30 '18 at 04:44
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1By the way here is a physics stack exchange answer for just that https://physics.stackexchange.com/questions/60299/difference-between-slanted-indices-on-a-tensor – Triatticus Jan 30 '18 at 04:49