So, I know that DNS is important in the running of CDN services. DNS allows the routing client traffic to the physically nearest CDN server. Question is, how can I do the same thing without any DNS involved? Note the edge servers are in different geographical locations.
Using one public IP over multiple locations are called Anycast and it is not related to DNS. DNS just translates FQDN to IP address.
If you assign same IP to different locations, your ISP will route the clients to the closer location. Is your application okay with this?
The most basic use case for anycast routing is hosting DNS servers. When you send DNS query to 126.96.36.199, you will get response from closest data center.
DNS involvement is not required for anycast. You should agree with your ISP if they allow this, what is their path selection criteria etc. If you have eBGP neighborship with your ISP from two locations, you can easily advertise same subnets. This is the common use case for geo-redundancy.