Reincarnation is not eternal life in the sense Jesus or any Christian talks about their resurrection. We do not come back as a different person living a separate life from the past person we were.
You want to know about "different person", there it is.
Except you'd probably consider it close to studying witchcraft I"d recommend you just read through the general beliefs about reincarnation among hindus (for example, they're big on this and have some very complex discussions of the matter) and reflect on what Christianity really tells you about RESURRECTION.
I just happened to see your side discussion about reincarnation.
Since I have studied the Hindu scriptures (Vedas and related texts) for more than 40 years, perhaps I can make a quick little description of reincarnation.
Very simply put, each living being is in reality an eternal soul, or spiritual spark, called jiva atma. Each soul or eternal self has no beginning and no end. When the soul or atma lives in the realm of birth and death (the material universe), he is imprisoned on the wheel of samsara - repeated birth and death, travlling from one body to another, as karma (actions taken by each individual and the subsequent reaction, either good or bad) and desires dictate.
The goal of life is to re-unite with God in love and become liberated from the constant repetition of birth and death. The Vedas describe an eternal effulgent realm beyond the confines of the material creation.
The bodily identities (male or female, one nationality or another, one race or another, one species or another) are not eternal and not who “we” really are. They are compared to a vehicle we are riding in, or a suit of clothes we are wearing.
So, the “person” we are now - say an American man, or a Chinese woman - is not our real self. It is only a temporary role; our real self is an eternal part of God. Not part in the sense of a drop of water is part of the ocean and again “becomes” the ocean (although no drop by itself is the whole ocean) but part of in the sense that a child is part of the father and mother; an eternal relationship, but eternal individuality and free will.