Induced maps on tensor spaces

[quasar_4]

Hi all,

Given a map P: V-->W for vector spaces V and W and the map P*: W* --> V* we have the relationship that many of us are familiar with:

For e in V, f in W, E in V* and F in W*, we can say that

(P*(F))(e)=F(P(e)).

This is nice and fine. So this is kind of the case for a rank 1 tensor. Now can anyone help me generalize this to rank r tensors, namely those of type (0,r) and (r,0)? We can't worry about the case of type (r,s) unless we know that P is an invertible map. But I'm having a REALLY REALLY hard time understanding the case for some higher rank tensor.

I am also hoping that whoever can help with this can introduce it with also explaining which of these induced maps is the pushforward and which is the pullback.

Thanks.

 

[llarsen]

To use the same notation as you have used here, it is convenient to represent a tensor as an r form using the covariant tensor: $$F_1 \otimes F_2 \otimes ... \otimes F_r$$. To map this to a real number (as you did with a rank 1 tensor above), we contract this with a contravariant tensor $$e_1 \otimes e_2 \otimes ... \otimes e_r$$. Extending this is actually rather simple, since each tensor term is simply mapped individually as follows:

$$[P^*(F_1 \otimes F_2 \otimes ... \otimes F_r)](e_1\otimes e_2 \otimes ... \otimes e_r) = [F_1 \otimes F_2 \otimes ... \otimes F_r](P(e_1\otimes e_2 \otimes ... \otimes e_r))$$

which can be rewritten as:

$$[P^*(F_1) \otimes P^*(F_2) \otimes ... \otimes P^*(F_r)](e_1\otimes e_2 \otimes ... \otimes e_r) = [F_1 \otimes F_2 \otimes ... \otimes F_r](P(e_1)\otimes P(e_2) \otimes ... \otimes P(e_r))$$

which reduces to the scalar product:

$$[P^*(F_1)](e_1) \cdot [P^*(F_2)](e_2) \cdot ... \cdot [P^*(F_r)](e_r) = F_1(P(e_1)) \cdot F_2(P(e_2)) \cdot ... \cdot F_r(P(e_r))$$

Covectors are mapped via pull back $$P^*(F) \in V^*$$ whereas vectors are mapped via push forward $$P(e) \in W$$.

 

[tacman]

Hello there,

Induced maps on tensor spaces can be a bit tricky to understand, but I will try my best to explain the generalization for higher rank tensors. First, let's go over the definitions of pushforward and pullback maps. A pushforward map is a linear map between two vector spaces that takes a vector in one space and maps it to a vector in the other space. On the other hand, a pullback map is a linear map between two dual spaces that takes a covector in one space and maps it to a covector in the other space. Now, let's see how these maps are related to induced maps on tensor spaces.

For a rank (0,r) tensor T in V*⊗...⊗V* (r times), the induced map by P would be P*: V*⊗...⊗V* --> W*⊗...⊗W* (r times), given by P*(T) = T∘P. This means that for a vector v1∈V, v2∈V, ..., vr∈V, and a covector f1∈W*, f2∈W*, ..., fr∈W*, we have (P*(T))(v1,v2,...,vr) = T(P(v1),P(v2),...,P(vr)) = T(f1,f2,...,fr) = (T∘P)(f1,f2,...,fr). This is the pushforward map, as it takes a tensor in the domain space and maps it to a tensor in the codomain space.

For a rank (r,0) tensor S in V⊗...⊗V (r times), the induced map by P would be P*: V⊗...⊗V --> W⊗...⊗W (r times), given by P*(S) = P∘S. This means that for a vector v1∈V, v2∈V, ..., vr∈V, and a covector f1∈W*, f2∈W*, ..., fr∈W*, we have (P*(S))(f1,f2,...,fr) = P(S(f1),S(f2),...,S(fr)) = P(v1,v2,...,vr) = (P∘S)(v1,v2