# Pushforward of covariant and contravariant tensor.

Le $$F : M \rightarrow N$$ be a map between manifolds. What is the pushforward of a covariant or contravariant tensor? I think that for a covariant tensor $$T : T_pM \times...\times T_pM \rightarrow \mathbb{R}$$ it is $$F_*T(X_1,...,X_n)=T(F_*X_1,...,F_*X_n)$$ but what for a contravariant tensor $$T : T_p^*M \times...\times T_p^*M \rightarrow \mathbb{R}$$? (note here $$X_1,...,X_n$$ are derivation of $$T_pN$$).

## 1 Answer

There seems to be some confusion of terminology here. In summary, given a smooth map $$F: M \to N$$ between manifolds:

• The differential of $$F$$ defines at each point $$p \in M$$ a pushforward $$(F_*)_p : T_p M \to T_{F(p)} N$$. (In general, the pushforward $$F_* X$$ of a vector-field on $$M$$ is not well defined.)
• If $$T : T_p M \times \cdots \times T_p M \to \Bbb R$$ is a (covariant) tensor, in general there is no such thing as a pushforward $$T_* F$$ on $$N$$. Instead, if $$S : TN \times \cdots \times TN \to \Bbb R$$ is a (covariant) tensor on $$N$$, then $$F$$ determines a pullback $$F^* S : TM \times \cdots \times TM \to \Bbb R$$ of $$N$$ to $$M$$ defined by $$(F^* S)_p (X_1, \ldots, X_k) = S_{F(p)} (F_* X_1, \ldots, F_* X_k)$$ (I suggest convincing yourself that both sides make sense).
• If $$T : T^*_p M \times \cdots \times T^*_p M \to \Bbb R$$ is a contravariant tensor, by dualizing we can identify it with an element of of $$T_p M \times \cdots \times T_p M$$. If $$T$$ is simple, so that we can write it as a sum of simple tensors $$\alpha_1 \otimes \cdots \otimes \alpha_k$$, $$\alpha_i \in T_p M$$, (it is rare to use this construction but) we can define the pushforward by $$F_* T := (F_* \alpha_1) \otimes \cdots \otimes (F_* \alpha_k) .$$ Then, we can define $$F_* T$$ for arbitrary contravariant tensors by linearity. As for vector fields, in general there is no such thing as a pushforward of a contravariant tensor field on $$M$$.
• thank you. I don't understand the meaning of $T_*X_1,...,T_*X_n$ though, shouldn't it be $F_*X_1,...,F_*X_n$? – roi_saumon Mar 5 at 22:04
• Yes, I've fixed the error, cheers! – Travis Mar 5 at 22:08
• (And you're welcome.) – Travis Mar 5 at 22:09