Pullback of tensors under general maps: The pullback of a covariant k -tensor α (one taking only contravariant vectors as arguments) under a map F : M → N is a linear map F ∗ ∗ : : T F ( p ) k N → → T p k M , where for any vector space V, T k V = V ∗ ∗ ⊗ ⊗ V ∗ ∗ ⊗ ⊗ ⋯ ⋯ ⊗ ⊗ V ∗ ∗ ⏟ ⏟ k times .