1a. Social tensions were manifested in the recent political crisis.
1b. The recent political crisis manifested social tensions.
1c. Social tensions were manifeste
in the recent political crisis.
1d. The recent political crisis was manifeste in social tensions.

Query: Which is/are grammatically correct?
Which is the best? Which is wrong?
Do they all have the same meaning?

Question

1a.  Social tensions were manifested in the recent political crisis.
1b.  The recent political crisis manifested social tensions.
1c.    Social tensions were manifeste
  in the recent political crisis.
1d.  The recent political crisis was  manifeste in social tensions.

Query:  Which is/are grammatically correct?
Which is the best? Which is wrong?
Do they all  have the same meaning?

Answer

As Dan said, all ok. But c) and d) need manifested not manifeste

Answer

1A is OK.
1B is OK.
1C is OK if you fix the typo - change manifeste (French word) to manifest (English word).
1D is OK if you fix the typo

They all have different meanings, except 1A & 1B mean the same as each other. 1C has a similar meaning to 1A & 1B.

1D doesn't make so much sense as the others, because it is saying the crisis is inside the tension, while the other three are saying the tension is inside the crisis which makes a bit more sense.

I would not personally use 1C or 1D because manifest as an adjective is very rare, so poorly educated people will not know what you are saying. Instead of saying "manifest", you could say "apparent" which means something very similar and is much more commonly used.

Answer

"Manifest" is both as an adjective and a verb.  Both uses are common.  Examples:

as a verb:
"The children manifested bad behavior".   ("manifested" = "displayed")

as an adjective:
"The children's bad behavior was manifest."   ("manifest" = "clearly visible")