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
Essa expressão pode ser reescrita sa seguinte forma: $$ \Xi = e^{2\psi} \nabla^M \psi f^{\ast}h + d\psi \otimes d\mathbb{I} + d\mathbb{I} \otimes d\psi, $$
onde \( \mathbb{I}: M \longrightarrow M \) denota a o mapa de identidade da base \( M \).
∇∗dΓf(X,Y) = (0, ̄∇df(X,Y) +df(Ξ(X,Y)))⊥
Decomposição da segunda forma fundamental em termos da hessiana da aplicação
(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))
Decomposição da conexão pullback, via aplicação gráfica
h(x) =e2ψ(x)h(f(x)) is a Riemannian metric on the pullback tangent bundlef−1TN
Vide definição de fibrado pullback
h(x) =e2ψ(x)h(f(x)) is a Riemannian metric on the pullback tangent bundlef−1TN
nduces onMthe graph metric (1.7),g∗(X,Y) =g(X,Y) + ̃h(x)(df(X),df(Y))
nduces onMthe graph metric (1.7),g∗(X,Y) =g(X,Y) + ̃h(x)(df(X),df(Y))
seen as the embedding ofMby the graph map Γf:M→ ̃M, Γf(x) = (x,f(x)),
f:M→N, and the graph submanifold, Γf={(x,f(x)) :x∈M} ⊂ ̃M
seen as the embedding ofMby the graph map Γf:M→ ̃M, Γf(x) = (x,f(x)),
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.