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Kronecker delta confusion

I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same): ...
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understanding the oscillating part of the Gutzwiller trace

given the density of states according to Gutzwiller's trace formula $ g(E)= g_{smooth}(E)+ g_{osc}(E) $ i know that the 'smooth' part comes from $ g_{smooth}(E)= \iint dxdp \delta(E-p^{2}-V(x)) $ ...