PERELMAN’S PROOF OF POINCAR ́E7in coordinates via theChristoffel symbolΓαβγ:=12gαδ(∂βgγδ+∂γgβδ−∂δgβγ).This in turn leads to theRiemann curvature tensorRiemδαβγ:=∂αΓδβγ−∂βΓδαγ+ ΓσβγΓδασ−ΓσαγΓδβγwhich then contracts to form theRicci curvature tensorRicαγ:= Riemβαβγwhich in turn contracts to form thescalar curvatureR:=gαγRicαγ.From the point of view of dimensional analysis16, it is helpful to keep in mind theschematic form of these tensors (thus omitting indices, signs, and constants):Γ =O(g−1∂g)Riem,Ric =O(g−1∂2g) +O(g−2(∂g)(∂g))R=O(g−2∂2g) +O(g−3(∂g)(∂ g)).In three spatial dimensions, one also hasdg=O(g3/2)dx. Of course, these schematicforms destroys all the geometric structure, which is absolutely essential for suchtasks as establishing monotonicity formulae, but for the purposes of analytical as-pects of the Ricci flow, such as the local existence theory, these schematic formsare often adequate.Given two orthonormal tangent vectorsX=Xα,Y=Yβat a pointx, thesectional curvatureK(X, Y) of the tangent plane spanned byXandYis given bythe formulaK(X, Y) = RiemδαβγXαYβXγYδ.One can show that this curvature depends only on the plane spanned byXandY,indeed it can be interpreted as the Gauss curvature of this plane atx. Schematically,Khas the same units as the Ricci scalar:K=O(g−2∂2g)+O(g−3(∂g)(∂g)). Indeedone can think ofR(up to an absolute constant) as the expected sectional curvatureof a randomly selected 2-plane, whereas the Ricci bilinear form RicαβXαXβfor aunit tangent vectorXis the expected sectional curvature of a randomly selected2-plane passing throughX.TheLaplace-Beltramioperator ∆gis defined on scalar functionsfas∆gf=gαβ∂α∂βf−gαβΓγαβ∂γfor schematically as∆gf=O(g−1∂2f) +O(g−2(∂g)(∂f)).It is worth noting that ∆ghas the same scaling as the Ricci scalarR. As such, theRicci scalar often naturally arises as a lower order correction term to the Laplacian.This is for instance evident in the formula for theconformal Laplacian∆g+18R