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By Chow B., Lu P., Ni L.

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By the deRham theorem, this implies (M n , g) is isometric to R × N n−1 , h , where N n−1 , h = x ∈ M n : bγ+ (x) = 0 . 78. In the study of the Ricci flow on 3-manifolds one of the primary singularity models is the round cylinder S 2 × R. This singularity model corresponds to neck pinching. A submanifold S ⊂ M n is totally convex if for every x, y ∈ S and any geodesic γ (not necessarily minimal) joining x and y we have γ ⊂ S. We say that S is totally geodesic if its second fundamental form is zero.

Next we look at the terms on the first two lines of the above equation. For 14 1. BASIC RIEMANNIAN GEOMETRY example the first terms on the top two lines are ∇ i ∇ k ∇ j X ℓ − ∇j ∇ k ∇ i X ℓ = ∇ i ∇ j ∇ k X ℓ − ∇j ∇ i ∇ k X ℓ − ∇i (Rkjℓm Xm ) + ∇j (Rkiℓm Xm ) = −Rijkm ∇m Xℓ − Rijℓm ∇k Xm − ∇i (Rkjℓm Xm ) + ∇j (Rkiℓm Xm ) Similarly (switch k and ℓ in the above) −∇i ∇ℓ ∇j Xk + ∇j ∇ℓ ∇i Xk = Rijℓm ∇m Xk + Rijkm ∇ℓ Xm + ∇i (Rℓjkm Xm ) − ∇j (Rℓikm Xm ) . Next we look at ∇i ∇k ∇ℓ Xj − ∇i ∇ℓ ∇k Xj = −∇i (Rkℓjm Xm ) and −∇j ∇k ∇ℓ Xi + ∇j ∇ℓ ∇k Xi = ∇j (Rkℓim Xm ) .

Yr )) − i=1 α (Y1 , . . , ∇X Yi , . . , Yr ) , where each term is an element of C ∞ (⊗s T M ). Let ⊗r,s M = (⊗r T M ∗ ) ⊗ (⊗s T M ) . The covariant derivative may be considered as: where or equivalently, ∇ : C ∞ (⊗r,s M ) → C ∞ (⊗r+1,s M ), ∇α (X, Z1 , . . , Zr ) ∇X α (Z1 , . . , Zr ) , n ∇α = i=1 ∇i α ⊗ dxi . 8 1. BASIC RIEMANNIAN GEOMETRY In this way we may square the covariant derivative operator: which is given by ∇2 : C ∞ (⊗r,s M ) → C ∞ (⊗r+2,s M ), ∇2 α (X, Y, Z1 , . . , Zr ) = ∇X (∇α) (Y, Z1 , .

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