"Manifest" vs. "manifested"
Which of the following is correct?
This idea is manifest in two influential theories.
This idea is manifested in two influential theories.
What I want to say is that there is a common idea underlying both theories.
Solution 1:
As Colin noted, 'manifest' works as an adjective, and 'manifested' as a past participle, in your example sentences. That 'manifest' connotes clearly visible has been mentioned in several comments, making 'manifest' unsuitable if manifestation of the idea in the theories is less than obvious.
As you seem to be seeking a fine-sounding way to say that two theories incorporate or are anchored upon the same idea, you might try 'immanent', an adjective meaning "naturally part of something; existing throughout and within something; inherent; integral; intrinsic; indwelling". For example, "This idea is immanent in two influential theories."
'Embody' (verb tr., "To include or represent, especially as part of a cohesive whole") also might work: "This idea is embodied in two influential theories."
'Intrinsic (adj., "innate, inherent, inseparable from the thing itself, essential") (or one of those synonyms, innate, inherent, inseparable, essential) would emphasize that the idea is critical: "This idea is an intrinsic part of two influential theories."
If the idea's influence on the theories is quite thoroughgoing, but not quintessential or foundational, then perhaps look at forms of permeate, pervade, saturate, imbue.
Solution 2:
"Manifest" is an adjective, meaning "clearly visible".
"Manifest" is also a verb, meaning "make visible", which has a past participle "manifested".
So both are grammatical, with hardly any difference in meaning. (Perhaps "manifested" would imply that somebody or something is doing the manifesting").
However, I'm not sure that either of them means what you want to say, because "manifest" has this sense of being clearly visible, not just being there. Perhaps "inherent" would be better?
Solution 3:
How about "This idea manifests in two influential theories?" I have a feeling the three formulations have subtle differences, but I can't put my finger on them.