5 Matching Annotations
1. May 2017
2. arxiv.org arxiv.org
1. (2q−1)!〈[T(2q−1)α, Aβ]·e⊤α, e⊤β〉

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3. arxiv.org arxiv.org
1. S⊥=hR⊥(∇fα,∇fβ)ηβ, ηαi

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4. arxiv.org arxiv.org
1. 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

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5. arxiv.org arxiv.org
1. by the Gauss formula we have(4.8)eRslij=hsihlj−hlihsj.
2. 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

Esse resultado faz uso apenas do fato de que o ambiente tem curvatura seccional constante, da fórmula de Gauss (vide nota anterior) e das definições do tensor de curvatura $$\tilde{P}_{(k)}$$ e do tensor de Newton, respectivamente.