A computation in coordinates shows that the Ricci tensor ofhis given byRich(X,X) =−(1V∆gV)h(X,X),Rich(X,Z) = 0,Rich(Y,Z) =Ricg(Y,Z)−1V(HessgV)(Y,Z)

The structure of the metrichnear the singular set clearly implies thatgeodesics realizing the distance between a point inNand a component of∂Mmeets∂Morthogonally. The proof of this fact is essentially the sameas the proof of the Gauss’ Lemma.

TM-valued symmetric bilinear form Ξ :TxM×TxM→TxM,Ξ(X,Y) = ̃h(df(X),df(Y))∇Mψ+g(∇Mψ,X)Y+g(∇Mψ,Y)X

(2q−1)!〈[T(2q−1)α, Aβ]·e⊤α, e⊤β〉

S⊥=hR⊥(∇fα,∇fβ)ηβ, ηαi

the 2-tensorE(k)is defined byE(k)ij:=−12k+1gliδli1i2···i2k−1i2kjj1j2···j2k−1i2kRi1i2j1j2···Ri2k−1i2kj2k−1j2k.Here the generalized Kronecker delta is defined byδj1j2...jri1i2...ir= detδj1i1δj2i1···δjri1δj1i2δj2i2···δjri2............δj1irδj2ir···δjrir.As a convention we setE(0)= 1. It is clear to see thatE(1)is the Einstein tensor. The tensorE(k)ijis a very natural generalization of the Einstein tensor. We callE(k)thek-th Lovelockcurvature

by the Gauss formula we have(4.8)eRslij=hsihlj−hlihsj.

Pstjl(k)=12δi1i2···i2k−3i2k−2stj1j2···j2k−3j2k−2j2k−1j2khj1i1hj2i2···hj2k−2i2k−2gj2k−1jgj2kl,which implies by (2.20) that(4.10)2ePstjl(k)hsj= (2k−1)! (T(2k−1))tpgpl

∇∗dΓf(X,Y) = (0, ̄∇df(X,Y) +df(Ξ(X,Y)))⊥

∇df(X,Y) =∇f−1X(df(Y))−df(∇MXY) =∇Ndf(X)(df(Y))−df(∇MXY)

Hessian offfor the Levi-Civita connections∇Mand∇N

(Y,U)⊤and (Y,U)⊥the ̃g-orthogonal projections ontoTΓfandNΓfrespec-tively

∇∗dΓf(X,Y) = ̃∇Γ−1fX(dΓf(Y))−dΓf(∇∗XY)

second fundamentalform∇∗dΓf:TM×TM→NΓfof Γf

∇∗be the Levi Civita connection ofMfor the graph metricg∗

̃∇Γ−1fX(Y,U) = (∇MXY,∇f−1XU) + (− ̃h(df(X),U)∇Mψ , dψ(X)U+dψ(Y)df(X))

h(x) =e2ψ(x)h(f(x)) is a Riemannian metric on the pullback tangent bundlef−1TN

h(x) =e2ψ(x)h(f(x)) is a Riemannian metric on the pullback tangent bundlef−1TN

Riemannian manifolds (Mm,g) and (Nn,h), and a functionψ:M→R,defining a Riemannian space ( ̃M, ̃g), where ̃M=M×Nand ̃g=g+e2ψh.