My professor recently defined immersions and embeddings in class, but he didn't really make any attempt to motivate these definitions. Now a few weeks later, after we've been using these concepts in class I'm still having trouble understanding what is *natural* about these definitions, what they *really mean* is still murky... I wonder if there's anyone here who can help me motivate these definitions? Here's the actual definitions used: Immersion: a function f from one manifold M to another N is an immersion if it's differential df_p, mapping from a tangent space T_pM to T_f(p)N is injective. Embedding: An immersion is an embedding if it is 1-1 and it's inverse is continuous.