In hindu mythology, many rivers are referred to as goddesses for example Ganga and Saraswati. Is there any curse given to them so that they all were in river form? What is the story behind it?
There are different stories related to different rivers but most of them land up in the mortal plane because of certain curses. According to the Brahma Vaivarta Purana Yamuna was born from Viraja, a cowherdess living in Goloka Vrindavan after a curse by Radha ji.
Ordered by Lord Krsna in Goloka, Yamuna, the best of rivers circumambulated the Lord and was about to go to earth. Then the Viraja river and the spiritual Ganga river both entered the Yamuna. For this reason the people know that the Yamuna is the greatest of rivers and the first queen of Lord Krsna.
Before I share more details about the other rivers I am sharing a map of ancient India with the rivers the way they were earlier:
Moving on from Yamuna, the Devi Bhagvatam Book 9 Chapter 6 also mentions that the reason for the descent of Ganga, Saraswati and Lakshmi (as Padmavati) in the form of rivers was due to a curse:
16-21. Nârâyana said :-- Hear, O Nârada! I will now describe that incident, the hearing of which removes all the sins. Laksmî, Sarasvatî and Gangâ, the three wives of Hari and all equally loved, remain always close to Hari. One day Gangâ cast side-long glances frequently towards Nârâyana and was eagerly looking at Him, with smile on Her lips. Seeing this, the Lord Nârâyana, startled and looked at Gangâ and smiled also. Laksmî saw that, but she did not take any offence. But Sarasvatî became very angry. Padmâ (Laksmî) who was of Sattva Guna, began to console in various ways the wrathful Sarasvatî; but she could not be appeased by any means. Rather Her face became red out of anger; she began to tremble out of her feelings (passion); Her lips quivered; and She began to speak to Her husband.
Lord Vishnu perhaps realizing that he should best remain out of a the women's argument left the place and then started the curses and counter curses. Saraswati tries to attack Ganga but Lakshmi intervenes so gets cursed by the former. Seeing this Ganga gets angry and curses Saraswati:
39-44. Thus saying, Gangâ became ready to curse Sarasvatî and addressing Laksmî, said :-- “O Dear Padme! As that woman has cursed you to become a river, so I too curse her, that she, too, be turned into a river and she would go to the abode of men, the sinners, to the world and take their heaps of sins.” Hearing this curse of Gangâ, Sarasvatî gave her curse, “You, too, will have to descend into the Bhurloka (the world) as a river, taking all the sins of the sinners.”
When all the action is over Lord Vishnu returns and tells them:
45-67. O Laksmî! Let you be born in parts, without being born in any womb, in the world as the daughter in the house of the King Dharma-dhvaja. You will have to take the form of a tree there, out of this evil turn of fate. There S'ankhachûda, the Indra of the Asuras, born of my parts will marry you. After that you will come back here and be my wife as now. There is no doubt in this. You will be named Tulasî, the purifier of the three worlds, in Bhârata. O Beautiful One! Now go there quickly and be a river in your parts under the name Padmâvatî.
O Gange! You will also have to take incarnation in Bhârata as a river, purifying all the worlds, to destroy the sins of the inhabitants of Bhârata. Bhagiratha will take you there after much entreating and worshipping you; and you will be famous by the name Bhagirathî, the most sanctifying river in the world. There, the Ocean born of my parts, and the King S'ântanu also born of my parts will be your husbands.O Bharatî! Let you go also and incarnate in part in Bhârata under the curse of Gangâ. O Good-natured One! Now go in full Amsas to Brahmâ and become His wife. Let Gangâ go also in Her fullness to S'iva. Let Padmâ remain with Me. Padmâ is of a peaceful nature, void of anger, devoted to Me and of a Sâttvika nature.
Chapter 8 of Book 9 also mentions the same:
Nârâyana said :-- A part of Sarasvatî descended in this Bhârata Punya Bhûmi (land of merits), owing to the curse of Gangâ; and She remained in full in Visnu's region, the abode of Vaikuntha. She is named Bhâratî, on account of Her coming to Bhârata; she is called Brâhmî because she is dear to Brahmâ; and She is called Vânî as She presides over Speech. Hari is seen everywhere, in tanks, in wells, in running streams (i.e., in Saras). Because He resides in Saras, therefore He is called Sarasvân. Vânî is the S'akti of that Sarasvân; therefore She is denominated Sarasvatî. The river Sarasvatî is a very sacred Tîrtha. She is the burning fire to the fuel of sins, of sinners.
O Nârada! Through the curse of Sarasvatî, the Devî Gangâ also assumed the form of a river in part. She was brought down to this earth at the request of Bhagîratha. Hence she is called Bhâgirathî. While Gangâ was rushing down to the earth S'iva capable to bear the great rush of Her, held Her on His head at the request of the Mother Earth.
Laksmî also, through the curse of Sarasvatî came in part of parts Bhârata as the river Padmâvatî. But She remained in full with Hari. Laksmî appeared also in Her other part as the well-known daughter Tulasî of the king Dharmadhvaja in India. Last of all, through Bhâratî's curse and by the command of S'rî Hari, she turned into the Tulasî tree, purifying the whole world.
Coming to the southern rivers, there are similar stories related to the coming down of rivers Krishna and Veni as well.
There are of course exceptions like the river Tapti who was Surya's daughter and came down to earth when she married an ancestor of the Pandavs known as Samvaran. You can check out the detailed story in SECTION CLXXIII of the Chaitraratha Parva of Mahabharat but I am giving the highlights here:
Tapati, the daughter of the god Vivaswat, was the younger sister of Savitri, and she was celebrated throughout the three worlds and devoted to ascetic penances. There was no woman amongst the celestials, the Asuras, the Yakshas, the Rakshasas, the Apsaras, and the Gandharvas, who was equal to her in beauty. Of perfect, symmetrical and faultless features, of black and large eyes, and in beautiful attire, the girl was chaste and of perfect conduct. And, O Bharata, seeing her Savitri (the sun) thought that there was none in the three worlds who, for his beauty, accomplishments, behaviour, and learning, deserved to be her husband. Beholding her attain the age of puberty and, therefore, worthy of being bestowed on a husband, her father knew no peace of mind, always thinking of the person he should select.
At that time, O son of Kunti, Riksha's son, that bull amongst the Kurus, the mighty king Samvarana, was duly worshipping Surya with offerings of Arghya and flower-garlands and scents, and with vows and fasts and ascetic penances of various kinds. Indeed, Samvarana was worshipping Surya constantly in all his glory, with devotion and humility and piety. And beholding Samvarana conversant with all rules of virtue and unequalled on earth for beauty, Surya regarded him as the fit husband for his daughter, Tapati. And, O thou of Kuru's race, Vivaswat then resolved to bestow his daughter on that best of kings, viz., Samvarana, the scion of a race of world-wide fame. As Surya himself in the heavens filleth the firmament with his splendour, so did king Samvarana on earth fill every region with the splendour of his good achievements.
River Godavari was formed when Ganga reappeared in the ashram of Gautam rishi to help absolve him of the sin of go-hatya as mentioned in the Kotirudra Samhita of the Shiva Purana. Similarly, according to the Skanda Puran, river Gomati was also a form of Ganga who appeared to help the Manas Putras complete their prayers:
There is an interesting tale describing how sage Vashishth had brought down Gomti from the heaven to earth. .... First of all Brahma created his ten manasputras and sought their help in increasing the population of the world by becoming householders. But, all of them were extremely virtuous and did not show any inclination towards getting married. Ultimately all of them went to the western coast and engaged themselves in austere penance. Their penance continued for a number of years after which 'Sudarshan chakra' appeared before them. As all of them looked up in bewilderment, they heard a heavenly voice -' O sons of Brahma! Very soon the almighty Vishnu is going to manifest himself. The 'Chakra' you are seeing is his. You all must perform the rituals of 'ardhya' in the name of Lord Vishnu to show your reverence towards him.'
All the Manasputras eulogized Sudarshan-Chakra with deep devotion. All of them wished that there were a holy river, with whose water they could perform the rituals of ardhya. But to their utter dismay there was no such holy river nearby. They then remembered Lord Brahma, who immediately understood what they desired. Lord Brahma instructed Ganga by saying-' Ganga! Go to the earth, where you would be known as Gomti. Sage Vashishth will lead you to your destination. Just follow him like a daughter follows her father.'
Finally, when sage Vashishth followed by Gomti reached their destination, all the Manasputras were delighted. They eulogized Ganga and expressed their gratitude to Vashishth for bringing Ganga on earth. All of them thanked Vashishth by saying-' Since you have brought Gomti on earth, you would be considered as her father.' They offered 'ardhya' to Gomti and subsequently eulogized Lord Vishnu. ..... This way, all the Manasputras were finally successful in their objective of offering ardhya to Lord Vishnu with the holy water of Gomti. At last, Gomti having fulfilled her mission for which she had descended down to Earth merged with the ocean.
To conclude, some of the popular river-goddesses exist on earth because of certain curses and there are a few who descended to help certain illustrious humans.
Sarasvati was the main river during the Vedic age. It used to flow from Himalaya to Run of Kuchh in Gujrat through Rajasthan. The Ghaggar river bed is the remnant of the once mighty river. At the beginning of the Epic age, the river started dying. When the mighty river, which was once the source of inspiration for generations of the Vedic poets was dying, the transformation happened. As the river once used to flood its banks, similarly Vedic poets urged that the Goddess should continue to inspire their mind and thought forever. That is why she is the Goddess of knowledge.