As we connected that way, do you feel it will even be appropriate for me to message her?

So in my previous post I mentioned how yesterday in Smoor Cafe of Lavelle Road area of our city Bangalore in India, I had a very long and great interaction with this French woman who is an expat doing an internship in the city. We talked a lot yesterday in the Cafe like she playfully asked me to guess her country, shared about the best French dishes, saying why Western France is the better than Paris to her, we talked about cars, hobbies and even career. She is 25 and I am also 25 so we are the same age and we both are freshly done with Masters but she did Masters in some other major and I did an MBA. So what happened is we connected on LinkedIn instead of Instagram or number. Yeah basically as we were talking about career, she also told me to follow the page of the office she is working in so I can apply for the Analytics roles I am looking for. So yeah I did and then I asked her whether we can connect on Instagram or Linkedin and she gave Linkedin instead and she connected back. It was such a great conversation and she was really very friendly and pretty. But what about next? Should I really message her on LinkedIn some day?

Unlike Instagram or Whatsapp, where casual or normal chats are fine, Linkedin is a professional networking app so yeah my message has to be really professional for real. So yeah I wonder how should I even message her on there some day. Like at least on Instagram I could have messaged her without worrying much like asking her "Hey how are you doing?" or just a whats up text but LinkedIn is more professional so just wondering what kind of starting message will be appropriate for the LinkedIn context plus I use LinkedIn to search for job openings and more professional networking. Or should I just leave it be without messaging? What do you feel? Give advice.