- An old man is sitting on the bench.
- There is an old man sitting on the bench.
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I'm sure if you wrote about an old man on a bench every day, at some point you would know him well enough, to simply call him "The old man". :)
An old man is sitting on the bench
There is an old man sitting on the bench
are perfectly equivalent. Use them interchangeably, picking the one whose rhythm you prefer in the context of the surrounding sentences.