isn't she too young to understand about hobbies at 3 years of age? you asked her if she has a hobby, you say she clearly said no and even then you took her to join bharatnatyam. i think more than her, you want her to learn it. i won't say you are wrong to have her learn somethign new, but it should definitely interest her too. and 3 is young for classical dance, at least the teachers who are affiliated to institutes such as gandharv will not teach children younger than 6 years of age.
I have heard that classical dancers start their training from as young as 3 or even 2 years, but I think unless you don't her to be a trained professional dancer one day, give her a year or more to begin. You can start at probably 4 or 5, and for now you can enrol her in some other activity class, if it is at all required. Else, she is just 3, let her have fun playing with her toys. Or choose some simple activity classes for her where they make things with dough or sand art and all.
hey i think 3 is a little too early for bharatnatyam. my elder one goes to bharatnatyam since she was 6 and the younger one also picked up, but when i enquired i was told that the youngest age to enroll is 6. teachers who abide by the gandharv or any other known institutes do not take younger kids as they feel they are not yet ready for all the instructions. you can try for western dance which will be more fun for her at this age
3 years is too young to learn bharatnatyam! even if the teacher says yes, the child will not be able to follow the basic instructions and mudras. i would suggest you take your child to an age appropriate place, where she can do colouring, learn some fun dance or any other activity such as swimming, that will interest her. as she grows up, she will understand what interests her, and you can accordingly enroll her there.