Amae is simply defined as the feeling of pleasurable dependence on another person but there is more to it. I'm including an example sentence for the sake of showing how the word can be used but this is not the only situation that amae can be observed.
In a romantic relationship, the woman will mostly feel _______ for the man who supports her.
Amae (甘え) is a word from Japanese; and it obviously has a cultural and historical significance in Japan (which is the usual case) but the concept is not unknown to Western cultures. It is often mentioned as untranslatable but here is an explanation from kirainet.com:
Amae (甘え) is a Japanese concept/word that is used to describe people’s behavior when you desire to be loved, you desire someone to take care of you, when you want unconsciously to be depending on another person (your parents, your wife/husband or even your boss) with a certain meaning of submission. For example, a person with lots of amae would be the one who is capricious so he/she gets the attention from other people, children are the best example of amae behavior, always aiming for pamper from their parents.
There are extensive studies on this concept and there are translations of this word considered as not a good equivalent. Japanese psychiatrist, Takeo Doi, claims that there isn't any equivalent of this term and it is unique to Japanese culture; but the following excerpt is from an article that compares amae in Japan and United States.
Doi defined Amae as the ability “to depend and presume upon another’s love or bask in another’s indulgence” (1992, p. 8) and called Amae “a key concept for the understanding not only of the psychological makeup of the individual Japanese but of the structure of Japanese society as a whole” (1973, p. 28).
Although Amae is a common word in the Japanese language, it has no exact equivalent in English. Some translations are “whining,” “sulking,” “coaxing,” “pouting,” “wheedling,” “being spoiled or pampered” (Johnson, 1993) and “cherishment” (Young-Bruehl & Bethelard, 2000), but none of these translations fully conveys the meaning of the complex phenomenon of Amae. For one thing, almost all of these terms have negative connotations in English, but Amae does not ordinarily elicit disapproval in Japan.
Doi took the lack of an English translation of the word Amae and the complexity of the concept as evidence that Amae is unique and central to Japanese culture, but he provided no empirical data to support this claim. The lack of English translation and the difficulty in defining the concept may be an indication that Amae is more salient and more frequently experienced in Japan, but they do not rule out the possibility that it exists in non-Japanese cultures.
Amae in Japan and the United States: An Exploration of a “Culturally Unique” Emotion by Yu Niiya and Phoebe C. Ellsworth (University of Michigan), Susumu Yamaguchi (University of Tokyo) / http://sitemaker.umich.edu
There is a lot to read and write about this concept but if we put the cultural differences aside, what would be an equivalent of amae in English and Western cultures?
Trust comes to mind but amae can be as strong as an indulgent love and as delicate as an infantile dependency. Thus, trust can be the basis of the amae relation but not the concept itself. Amae satisfies the subconscious desire for unconditional acceptance in a positive way but it is doesn't indicate submission alone.