# Download A Short Course in Differential Geometry and Topology by A. T. Fomenko PDF

By A. T. Fomenko

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15) then yield E E h = (π∗ E) − A∗E h c˙v − A∗c˙h E v + σ(c˙v , E v ), v = Ac˙h E h − SE h c˙v + E v v . 16) provides another proof of the fact that F splits locally if A and S are identically zero. One can argue this using either horizontal or vertical ﬁelds: for example, if E is a parallel ﬁeld along c with E(t0 ) vertical, then E v is parallel, so E = E v is always vertical. Thus, V is invariant under parallel translation, and the claim follows from de Rham’s holonomy theorem. 16) applied to the tangent ﬁeld c˙ of a curve c in M yields ˙ − 2A∗c˙h c˙v + σ(c˙v , c˙v ), c˙ h = (π∗ c) c˙ v = −Sc˙h c˙v + c˙v v .

Next, set V1 (t) := {J ∈ V | J(t) = 0}, V2 (t) := {J (t) | J ∈ V, J(t) = 0}. The map V1 (t) → V2 (t) which sends J ∈ V1 (t) to J (t) is linear, surjective by deﬁnition, and has zero kernel. Thus V1 (t) ∼ = V2 (t). For any ﬁxed t0 ∈ R, let J1 , . . , Jk be a basis of V1 (t0 ), and extend it to a basis J1 , . . , Jl of V, where l := dim V. Then Jk+1 (t0 ), . . , Jl (t0 ) is a basis of {J(t0 ) | J ∈ V}, and dim V = l = k + l − k = dim V1 (t0 ) + dim{J(t0 ) | J ∈ V} = dim V2 (t0 ) + dim{J(t0 ) | J ∈ V} = dim V (t0 ).

Jl of V, where l := dim V. Then Jk+1 (t0 ), . . , Jl (t0 ) is a basis of {J(t0 ) | J ∈ V}, and dim V = l = k + l − k = dim V1 (t0 ) + dim{J(t0 ) | J ∈ V} = dim V2 (t0 ) + dim{J(t0 ) | J ∈ V} = dim V (t0 ). To prove the second statement, let t0 and J1 , . . , Jl be as above. We will show that there is some > 0 such that J1 (t), . . , Jl (t) are linearly independent for each t ∈ (t0 , t0 + ). By assumption, Jk+1 (t0 ), . . , Jl (t0 ) are linearly independent, and thus remain so for t close enough to t0 .