Sure, here are the rewritten sentences:

1. They say he is able to swim across this river. (**swimming**)
He **is said to be capable of swimming across this river**.

2. They may be there at the meeting, or they may not. (**out**)
It is **up in the air whether they will be at the meeting or not**.

3. “Don’t forget to hand in the paper by the deadline,” said the teacher. (**reminder**)
The teacher **gave a reminder to hand in the paper by the deadline**.

4. Learning English is becoming more and more popular in our city. (**increasingly**)
It has **become increasingly popular to learn English in our city**.

5. Right after disembarkation from the plane, he was taken into custody. (**got**)
As **soon as he got off the plane, he was taken into custody**.

6. The teacher and his students knew nothing about the crack on the wall. (**idea**)
Neither **the teacher nor his students had any idea about the crack on the wall**.

7. The boy’s effort really impressed the audience and judges. (**left**)
It was **the boy’s effort that left a strong impression on the audience and judges**.

8. Instead, you must try to have a good relationship with the others. (**get**)
I wish you **would get along well with the others**.

9. I always go jogging in the morning to keep fit. (**invariably**)
In **the morning, I invariably go jogging to keep fit**.

10. Doubtlessly, Katie is the best violinist in the conservatory. (**no-one**)
It goes **without saying that no-one in the conservatory is a better violinist than Katie**.