To connect a Spring Boot application to YugabyteDB Managed and deploy it on Amazon Elastic Kubernetes Service (EKS), start by creating a free YugabyteDB Managed account and database cluster, then download the Spring Boot PetClinic sample application and connect it to your YugabyteDB cluster. Next, containerize the application using Docker and tag it with your ECR repository, then push the image to your repo in ECR. Deploy the Spring Boot PetClinic sample application image to EKS by creating a CloudFormation stack, updating your `kubectl` configuration to connect to your EKS cluster, and creating a new file named `manifest-eks.yml`. Finally, create the Service and Deployment on your EKS cluster using the `kubectl create -f manifest-eks.yml` command. The PetClinic sample application should now be available at the External IP address listed in the output of the `kubectl get svc` command.