Behavior of tensors under inclusion: For inclusion maps from a submanifold S into M and a covariant tensor α of order k on M it holds that ι ι ∗ ∗ α α ( X 1 , X 2 , … … , X k ) = α α ( ι ι ∗ ∗ X 1 , ι ι ∗ ∗ X 2 , … … , ι ι ∗ ∗ X k ) = α α ( X 1 , X 2 , … … , X k ) , where X 1, X 1, …, X k are vector fields on S. The subscript star denotes the pushforward (to be introduced later), and it is in this special case simply the identity map (as is the inclusion map). The latter equality holds because a tangent space to a submanifold at a point is in a canonical way a subspace of the tangent space of the manifold itself at the point in question. One may simply write ι ι ∗ ∗ α α = α α | S , meaning (with slight abuse of notation) the restriction of α to accept as input vectors tangent to some s ∈ S only.