After assassination, India turns to Rajiv

India has rushed to try to fill the tremendous leadership vacuum caused by the assassination of its formidable prime minister, Indira Gandhi. Mrs. Gandhi was gunned down early Wednesday by a fusillade of bullets from members of her own security guard in the lush gardens of the prime minister's residence.

Within hours, her only surviving son, Rajiv, was sworn in as successor to the world's longest-serving, democratically elected premier. His selection as the country's seventh prime minister apparently continues the extraordinary Nehru-Gandhi dynasty which has ruled this vast and complex land, except for two brief periods, throughout its 37 years of independence.

Despite outbreaks of rioting here in the capital and in at least six other Indian states, initial signs were that, politically, the abrupt transfer of power would proceed smoothly as in the past. All of the nation's most distinguished leaders - Mohandas Gandhi (no relation), Jawaharlal Nehru (Mrs. Gandhi's father), Lal Bahadur Shastri, and now Mrs. Gandhi herself - have died at the zenith of power; and the democratic fabric of the world's largest democracy has always held.

But Rajiv Gandhi faces challenges of daunting proportions, both in holding on to power and in keeping this disparate nation together. Though he was widely considered Mrs. Gandhi's chosen successor, his prompt installation came as something of a surprise. Only three weeks ago he told this correspondent that he was not yet ready to even think of following in his mother's controversial and charismatic footsteps. The 40-year-old former airline pilot and junior member of Parliament readily acknowledged that he was inexperienced and young.

His mother, on the other hand, behind the soft exterior and disarming smile, was an extraordinarily determined woman. As prime minister, and earlier as hostess for her father for 17 years, she saw the nation through three wars, won four elections nearly single-handedly, and was about to embark on what would almost certainly have been an unprecedented fifth term.

She was almost universally considered to have one of the surest political instincts in the business, until early this year when her legendary skill began, discernibly, to slip. Dispatching the Indian Army against rebellious Sikhs in their holiest shrine, the Golden Temple, and unseating three democratically elected state governments in as many months, had badly tarnished Indira Gandhi's image, both at home and abroad.

It was her direct challenge to the Sikh extremists which seems to have precipitated her assassination. Her assailants were both Sikhs.

According to the British actor-director Peter Ustinov, who was waiting for Mrs. Gandhi in her garden to do a television interview, the prime minister was ambushed shortly after she stepped out of her house.

''At first three shots were fired, as though a signal,'' said Mr. Ustinov, who heard but did not see the assassination. ''Then there was a burst of automatic fire.''

According to Mrs. Gandhi's information adviser, Sharada Prasad, the prime minister was approaching a gate linking two bungalows within the official complex when one of her bodyguards, Beant Singh, shot her with his service revolver. A second guard, Satwant Singh, then emptied the contents of his automatic rifle into the body.

Mrs. Gandhi was rushed to a hospital and an emergency operation was performed , but she died at 2:30 Wednesday afternoon.

Mr. Prasad said one of the two assassins, Satwant Singh, had been on the prime minister's residential security staff for eight years. He was overpowered by other security men but not killed and later reported ''out of danger'' in a hospital. He was from Gurdaspur district of the Punjab - a stronghold of the militant Sikh leader, Sant Jarnail Singh Bhindranwale, who was killed in the Army assault on the Golden Temple last June - and one of the Punjab's three districts still under Army control.

The other assassin, security sub-inspector Beant Singh, had joined the prime minister's security staff in January 1982. He was killed during the attack.

Security surrounding the prime minister had been tripled since the Army stormed the Golden Temple in Amritsar to dislodge Jarnail Singh Bhindranwale and some 400 hardcore followers encamped inside.

Since then, the prime minister had received a number of threats on her life. Most recent were official government reports on the weekend that four heavily armed, alleged Sikh extremists had been arrested while clandestinely crossing into India from neighbouring Pakistan. According to Bua Singh, Amritsar's district chief of police, they confessed under interrogation that they had been trained by Pakistani military officers, and instructed to kill a number of top Indian government officials - Mrs. Gandhi was not directly named.

The allegation was immediately, and categorically, denied by Pakistan. And Western diplomatic officials, long accustomed to charges and counter-charges emanating from both sides of the highly charged frontier, gave little initial credence to allegations of Pakistani complicity. They had long been concerned however about threats to the prime minister's life, by the nation's largely embittered community of 15 million Sikhs.

From the Golden Temple, Sikhdom's five high priests expressed their ''shock and deep grief'' at Mrs. Gandhi's death. They condemned cults of violence, and appealed to Sikhs and Hindus alike to maintain ''goodwill, affection, and peace'' at this critical period in Indian history.

The Sikhs' only political party, the Akali Dal, likewise condemned the assassination, and called on everyone to remain calm.

As the news swept through India, people responded in a state of shock. Throughout New Delhi, iron shutters banged down on the markets' shops. By late afternoon Wednesday, this normally vibrant and chaotic capital wore a deserted look.

Sharpshooters from the elite paramilitary force were positioned around all shrines and places of worship. Security was reinforced not only at the prime minister's official residence, but at all homes of government dignitaries whom intelligence sources identified as being on a ''terrorists' hit list.'' All intersections were heavily guarded by steel-helmeted security officials, their automatic weapons cocked.

Several people were seriously injured when a frenzied mob went on a rampage in downtown New Delhi, burning buses and private vehicles and dragging Sikh drivers and passengers out of cars. They shouted slogans against ''Khalistan'' - the Sikhs' visionary independent nation - and against Pakistan. Rioting by outraged and grieving Hindus was also reported from at least six other Indian states Wednesday night.

Meanwhile, the stunned nation began a 12-day period of official mourning Wednesday night. Mrs. Gandhi's body is to lie in state at her official residence through Friday, and be cremated on Saturday afternoon.

It was announced in Washington that Secretary of State George Shultz would represent the United States at the funeral. President Reagan expressed his condolences to India's ambassador to the US saying, ''We have lost a friend.'' The President issued a statement declaring his ''shock, revulsion, and grief over the brutal assassination.''

Earlier this week, on Tuesday, the tiny, artistocratic woman who ruled this nation of 700 million for nearly 16 years, wound-up a two-day visit to the state of Orissa and told a huge public meeting at Bhubaneswar's parade ground:

''If I die today, every drop of my blood will invigorate the nation .... I am not afraid of these things. I don't mind if my life goes in the service of the nation. That has been my life.''

During that visit to Orissa, where she inaugurated a $20 million air defense and missile training centee for the artillery regiment at Gorabandar, a woman sought to reach her with a petition, pushing her way through the crowd. She was stopped by Mrs. Gandhi's security guards. But the prime minister looked up and said, ''let the woman through.''

She also spoke of the grave threat to India's security, and the sub-continent's escalating arms race. She didn't have to mention Pakistan. Everyone present knew what she meant. The enlisted men cheered her wildly.

She smiled and added, ''in case of any eventuality, we will bury our differences and fight unitedly, as we did in the past, to preserve the country's unity and integrity.''

She smiled again and accepted a replica of a missile from Lt. General Bhupindar Singh, the Sikh army commander of the central command. It was Indira Gandhi's last photograph.