"This book belongs to me."
It is a "stressed" construction, meaning "this book is mine".
Therefore, depending on the emphasis you want to give to the fact the book is yours (more or less factual, more or less emphatic), you can translate "this book belongs to me" (and English variants) by:
- c'est mon livre
- ce livre m'appartient
- ce livre est à moi
- ce livre est le mien
In English, you express possession in two forms with verb BE:
- "this book is Peter's " = possessive case ('s) with a noun/name
- "this book is mine", with verb Be + possessive pronoun (yours)
The nearest translation in French with verb "ÊTRE" is: être + à + owner:
- ce livre est à Pierre (possessive case does not exist in French)
When the owner is represented by a personal pronoun, you have to use "stressed" pronouns: moi, toi, lui, elle, nous, vous, eux;
- ce livre est à moi / toi...