Tensors

There are several approaches to define tensors.

...

Anyway, given $U$ an open set of a manifold $M$, there are two defining properties for a map

$$ S:\mathfrak{X}(U)\times \mathfrak{X}(U)\times \mathfrak{X}(U)\times \cdots\to \mathfrak{X}(U)\times\mathfrak{X}(U)\times \cdots $$

to be a tensor:

The output only depends on the input vectors evaluated at a point, that is, we can create a map

$$ S_p:T_pM\times T_p M\times \cdots \to T_pM\times T_p M\times \cdots $$

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Author of the notes: Antonio J. Pan-Collantes

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