The Chakra (an Akula II) is the Indian Navy.
The next one, built to a similar standard, will also be "Indian Navy".
Outside of the Sea Wolf, Virginia, Yasen and Astute classes...there is no sub that can challenge them. They will present a very, very difficult challenge to any maritime nation wanting to transit the IO area.
Regarding the Akula, it's a lease to familiarize Indian Navy with operating nuclear submarines. It's partially manned by Russian instructors. The term of the lease, from what I understand, is that there are constraints on how it may be used. It cannot be considered a fully Indian naval asset. India has not purchased them but only reserves the option to do so once the lease expires.
They have an operational carrier, with an operational airwing of Mig-29Ks. They have a strong nuclear attack submarine, a strong air defense DDG, and strong Shivilak and Talwar FFGs to defend her. They will soon have a second such carrier group...and will build their third.
I do not agree at all that they are falling "rapidly behind"
Not in the least IMHO.
The carrier, as with any military asset, must be measured against the capabilities of potential opponents. A Nimitz might be formidable against Iran or Syria, but when going against a technologically advanced foe, it becomes more of a liability, having neither the number of planes necessary to secure air superiority nor the defenses necessary to deflect harm. Naturally, geographical advantages play a large part when India has her carriers prowling her backyard, but these ships are by no means difficult to target or sink. All carriers in this day and age are exceptionally vulnerable, provided the opponents have the technological means. India's carriers are far less potent, once we look past the symbolic aura of having a 'carrier'.
In terms of a transit across the IO, any nation will have a difficult chore of forcing their way past the Indian Navy and Naval Air Arm, and their Air Force even now. The Vikramaditya and the coming Vikrant are not qualitatively behind. The Shivalik, Talwar and Kolkata vessels are also not outclassed in any major way qualitatively if it came to a confrontation in the IO. The Akula SSNs are arguably ahead of any other maritime power outside the US, Russia and the UK, and even they would have to be wary of them.
These conditions will only grow in the years to come as the Indians add their third carrier, the Bravo Kolkatas, and their new ASW FFGs, as well as field their own SSNs in some numbers probably starting in the 2020s. I would not cavalierly dismiss them in the least and I can tell you the planners of all of the nations we have mentioned, as well as those we have not, are not doing so.
Yes Jeff, I agree India's geographic proximity to the vital sea lanes counts as a large advantage. That said, India's ability to patrol and monitor her near seas remains limited, not the least because she has fallen far behind her own acquisition goals and therefore lacked the necessary assets. Moreover, should the Chinese decide to set up basing agreements with countries around the Indian Ocean, India's geographic advantage will be severely reduced. This is not to say that China's military bases would be set up with India in mind, for China doesn't appear to consider India a military or strategic threat.
In the next twenty years, it's not beyond the realm of possibility that China would have naval and airbases dotted around the Indian Ocean, much as we do today. It's a natural progression as China's growing commercial ties with that part of the world necessitate the need to protect her commercial interests. The U.S. navy, if we are not bankrupt by then, cannot be trusted to do the job.
Jeff, I have no doubt that India will add new pieces to her navy in the coming years, but you should take India's projected acquisitions with a huge grain of salt. India has always over-promised and under-delivered by a wide, wide margin. The lead ship of P15A was supposed to see service in 2007, and we all know what actually transpired. These kinds of costly delays are the norm, Jeff, and almost never the exception. I'd hold off on predicting what units will be added to the Indian Navy until they are actually added. Meanwhile, a certain 'rival' up North are not having the same acquisition problems. The trend lines have been very clear for some time now, and it's not about to end any time soon.
And we haven't even touched on how India's lack of indigenous weapons industry severely holds her back strategically. One simply cannot build a strong military on imported hardware, self-evidently.