It's common for cats to lie down on their bellies when they see their owners, usually indicating that the cat feels safe and trusting. When a cat exposes its belly, it's one of its most vulnerable areas. If a cat is willing to do this in front of its owner, it generally means that it has a high level of trust in its owner and feels safe in its environment.
Lying on its back is a relaxed posture, indicating that the cat feels comfortable and content in its current environment. Some cats may use this posture to invite their owners to play or interact with them.
