She was physically distant at a date
I met a girl a while ago. That day, we had a nice conversation. A thing that struck me is that she talked a lot about herself and didn't show any interest in me. She really enjoyed chatting, but it was only about her. When we left, I offered to shake her hand. Even though she enjoyed my company, she was very hesitant and timid to shake my hand.
After a while, I had my first date with her. At the meeting point, I was about to at least shake her hand or give her a little hug, but she was physically very hesitant to come near me. She walked around me and got into my car, which really surprised me. Not even a handshake. In my car, she started to talk again, about her things. She never asked me anything, nor did she ever call me by my name, ever. We had long, interesting conversations about our mutual interests. She shares a lot of the same interests, which is why we can have endless conversations.
During the date, I did touch her arm, back and layed my hand on her knee. She was very relaxed about it and didn't reject it, but she didn't actively participate. We had a fun date and I paid for everything. She didn't offer to pay, nor did she thank me. When I was telling her something about myself, she looked careless and was hearing me out. When I stopped talking, she ignored what I just said and started talking about herself again. She did tell me that she had a good time. When I brought her home, she went out of my car. No hug, no kiss on the cheek, not even a handshake. She simply said she enjoyed it and want to go out again. And she still hasn't even used my name or asked me anything about me, as if she doesn't want to know.
I really like to hang out with this girl because she's nice and I'm very attracted to her, but I feel like it's not mutual. What should I do? Should I go have a second date? I know I don't have much to lose, but I don't want to be taken advantage of either.