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We use contraction of indices method to manipulate Tensors. However, I cannot relate that manipulation visually. We can change covariant tensor to contravariant tensor and vice versa by contracting indices with the metic tensor. Covariant tensor represent gradient and contravariant tensor basically represent tangent vector. So, using contraction we are doing something awesome but how exactly can I can see this visually. What exactly does contraction do geometrically?

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This link contains some pictures which you may sound useful. The stuff relevant to contraction is in the section entitled "transvection" - not a piece of terminology that you hear very often ! Basically vectors are represented by arrows and one forms are represented by families of parallel planes. The contraction of a vector with a form is the number which represents the number of planes that the vector pierces.

Also I'd recommend looking at Misner, Thorne and Wheeler's classic book as it contains a very pictorial presentation of this subject.

In the OP I think you meant to say that

We can change covariant tensor to contravariant tensor and vice versa by contracting indices with the metric tensor

(my bold)

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    $\begingroup$ The URL is dead :( $\endgroup$ Commented Apr 8, 2022 at 12:32
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    $\begingroup$ This particular dead URL has been rescued by the Internet Archive's Wayback Machine. $\endgroup$
    – rob
    Commented Apr 8, 2022 at 14:01

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