Proper piano technique does not stop at curved fingers.
Curved fingers are just one facet of proper piano playing technique. Other facets include but are not limited to:
1. Good, strong posture. You need to be centred at the piano. Your shoulders and arms should be strong but not tense, you should get power and strength from your core and your arms. Where you sit in front of the piano is equally as important, too close and you limit your arm movements and become stiff, too far away and you will feel uncomfortable and your movements will not be controlled.
2. Correct fingering. If my students had a dollar for every time I yelled 'FINGERING!' in a lesson, they would all be very rich. Using correct fingering is so crucial to playing piano with proper technique. With correct fingering you won't 'run out' of fingers in long, fast passages. With correct fingering you won't find your fingers doing any sort of weird acrobatic movements in order to reach notes, with correct fingering, your playing will improve and become more agile and smooth.
3. Consistency. This one also comes back to correct fingering. If there is a passage in your music that is repeated, use the same fingers to play it! Try to use similar gestures, move your hand and wrist the same way you did before.
What can you do to improve your technical abilities?
1. Be conscious of your own movements. Become more aware of your fingers as you play, don't just try to 'get the notes right'. Think about the smaller movements of your hands, glance down as you play to notice whether your pinky finger is curved or flat, whether your wrists are high and tense, low and floppy or flat and relaxed. Every now and then, think about your body, are you hunched over or sitting straight? Are your arms tense or relaxed? Are you sitting too close/too far away from the piano? Do you hold any tension in your body when you play or does your body feel light?
2. Listen to your teacher! This goes without saying. Your teacher will be giving you tips and ideas, ways to improve your playing or habits for you to be aware of and to try and fix. Take these things on board and follow the advice in number 1 above for when you practise post lesson.
3. Do technical exercises. Hands down the best way to improve your finger independence and technique is to do scales and technical exercises. I have a preference for Czerny, but I also like Hanon. If you can get your hands on Hanon book 1 that is a great place to start. For Czerny, please see this link: https://pianoexercises.org/exercises/czerny/
My favourites from this list and the ones that I give to my students are:
Practical Finger Exercises (this one is the best!)
125 Exercises in Passage Playing
25 Exercises for Small Hands
4. Practice makes progress. Finally, nothing will improve if you don't continue to practise consistently. "Practice makes progress" is what I tell myself and my students all the time. It is very true and it is the only real way to see consistent improvements in your playing. For ideas on how to practise when you are time poor, please see my blog post on this topic linked here.