Why does Isabel Archer in Portrait of a Lady go back to Gilbert Osmond in the end?
You have asked a question that has been debated for many years with no definite answer! We can tell only so much from the book and from there on in we have to surmise and infer Isabel's reasons for returning to her marriage with Gilbert Osmond. What we do know for sure is that her embrace with Caspar Goodwood was the trigger that propelled her back to Rome: there is "a very straight path" for her.
The main ideas therefore are as follows:
1) Some critics argue that Isabel Archel is not a tragic character, although she is in a bad situation. She resolves at the end of the novel to bring her full character and resources to bear on her marriage and her duty to her husband and step daughter. She therefore is rejecting the ways of the corrupt European circle that Gilbert Osmond and Madame Merle are part of and takes a more principled stand.
2) Some critics argue that Isabel returns to fulfil her duty because she does not want to end up with the same "empty independence" of her aunt that she has seen through being her protoge.
3) Others argue that Isabel is afraid of public shame and disgrace - public awareness of her mistake in marrying Gilbert Osmond. She made a vow of marriage to him that she will carry out, whatever the price.
4) My own personal view is that Isabel throughout the novel has displayed a fierce independence and determination, which says a lot about her strength of character. For her to run off with Gilbert Osmond would be the "easy" path - she could do it, but she is so resolute a character that I don't think she would have made this choice - picking the harder path is entirely in her character, and makes her all the more admirable and a heroine that lingers on in your mind long after you have finished the book. It appears that there are some choices in life where once you have made your choice you have to stick to it.