Hi I have trouble in the battery of my 2001 toyota vitz,the problem is when ever I connect the battery the rear break light glows and never goes off even if I start the engine and start driving and also the break indicator light glows even if I released the hand break fully.Do you think I have got the battery terminal connection wrong.

No there is a brake switch to turn on the brake lights when you step on the brake pedal. It is located near the rake pedal under the dash. It has more then likely failed.

The engine switches off on its own after running for a few minutes and the battery sign shows red on the dash board of my VITZ

If the battery was in wrong, the car would not start or drive.
The problem might be a faulty brake light switch.
also make sure the brake cylnder resivoir is full of brake fluid, and the brake pedal moves properly and the brakes are working.
Check the floor mat hasnt accidently moved up behind the brake lever, jamming the switch