Can I get a permanent tattoo of mother Earth as a respect towards her? I want to get it on my back just between the shoulders. Is getting a permanent tattoo on our body sacred according to Hinduism?
No, getting tattoo or human branding is not a part of original Hinduism.
The origin may be the ancient treatment of a slave (often without legal rights) as livestock.
European, American and other colonial slavers branded millions of slaves during the period of trans-Atlantic enslavement. Sometimes there were several brandings, e.g. for the Portuguese crown and the (consecutive) private owner(s), an extra cross after baptism as well as by African slave catchers. To a slave owner it would be logical to mark such property just like cattle, more so since humans are more able to escape.
In criminal law, branding with a hot iron was a mode of punishment consisting of marking the subject as if goods or animals, sometimes concurrently with their reduction of status in life.
Brand marks have also been used as a punishment for convicted criminals, combining physical punishment, as burns are very painful, with public humiliation (greatest if marked on a normally visible part of the body) which is here the more important intention, and with the imposition of an indelible criminal record. Robbers, like runaway slaves, were marked by the Romans with the letter F (fur); and the toilers in the mines, and convicts condemned to figure in gladiatorial shows, were branded on the forehead for identification. Under Constantine I the face was not permitted to be so disfigured, the branding being on the hand, arm or calf.
The Acts of Sharbil record it applied, amongst other tortures, to a Christian between the eyes and on the cheeks in Parthian Edessa at the time of the Roman Emperor Trajan on a judge's order for refusal to sacrifice.In the 16th century, German Anabaptists were branded with a cross on their foreheads for refusing to recant their faith and join the Roman Catholic church. In the North-American Puritan settlements of the 17th century, men and women sentenced for adultery were branded with an "A" letter on their chest, and for other crimes, such as "D" for drunkenness and "B" for blasphemy. Similar human branding is also shown in the movie Inferno-2016.