Return to Answer

The isometric immersion that you describe above is the higher dimensional pseudosphere. Now, concerning your final question, I presume that you need to search about isometric immersions of the hyperbolic space $\mathbb H^n$ by means of a warped product representation (of $\mathbb H^n$) into the Euclidean space.

Now, some additional things that you might be interested to:

1. There are many (explicit in some cases) local isometric immersions from $\mathbb H^n$ to $\mathbb R^{2n-1}$. These can be constructed by using either the Ribaucour or the Bäcklund transformation (for instance, see the papers by Dajczer-Tojeiro and Tenenblat-Terng).
2. Local isometric immersions of the hyperbolic plane $\mathbb H^2$ into $\mathbb R^3$ imply "local" solutions, that is, solutions that are not defined on the whole $\mathbb R^2$, of the sine-Gordon equation and vice versa. Therefore, it follows from Hilbert's theorem that there is no "global" solution, that is, a solution defined on the whole plane $\mathbb R^2$, of the sine-Gordon equation. Just like in the case of dimension two, the same also happens in the higher dimensional case where now you will end up with a system of PDES (see for instance Dajczer-Tojeiro). We can have local solutions to this system but we don't know if there exists any global. The existence of a global solution would imply the existence of a global isometric immersion of $\mathbb H^n$ into $\mathbb R^{2n-1}$, which would give a non affirmative answer to the major still open problem (in submanifolds) up to this day, which is the following conjectured extension of Hilbert's theorem:

There is no global isometric immersion from $\mathbb H^n$ to $\mathbb R^{2n-1}$

However, the above holds true in some very special cases. For instance:

• If the immersion is also minimal (the mean curvature vanishes) (see Moore).

I should also mention here that $\mathbb H^2$ admits no minimal immersion in any Euclidean space. (for a proof of this fact see either "Lectures on minimal submanifolds" by Lawson, or Bryant, or Di Scala).

• (weaker) If the immersion has also bounded mean curvature (see here)
• (even weaker) If also the length of the mean curvature of the immersion does not go to infinity too fast, that is, exponentially fast (see here)

The isometric immersion that you describe above is the higher dimensional pseudosphere. Now, concerning your final question, I presume that you need to search about isometric immersions of the hyperbolic space $\mathbb H^n$ by means of a warped product representation (of $\mathbb H^n$) into the Euclidean space.

Now, some additional things that you might be interested to:

1. There are many (explicit in some cases) local isometric immersions from $\mathbb H^n$ to $\mathbb R^{2n-1}$. These can be constructed by using either the Ribaucour or the Bäcklund transformation (for instance, see the papers by Dajczer-Tojeiro and Tenenblat-Terng).
2. Local isometric immersions of the hyperbolic plane $\mathbb H^2$ into $\mathbb R^3$ imply "local" solutions, that is, solutions that are not defined on the whole $\mathbb R^2$, of the sine-Gordon equation and vice versa. Therefore, it follows from Hilbert's theorem that there is no "global" solution, that is, a solution defined on the whole plane $\mathbb R^2$, of the sine-Gordon equation. Just like in the case of dimension two, the same also happens in the higher dimensional case where now you will end up with a system of PDES (see for instance Dajczer-Tojeiro). We can have local solutions to this system but we don't know if there exists any global. The existence of a global solution would imply the existence of a global isometric immersion of $\mathbb H^n$ into $\mathbb R^{2n-1}$, which would give a non affirmative answer to the major still open problem (in submanifolds) up to this day, which is the following conjectured extension of Hilbert's theorem:

There is no global isometric immersion from $\mathbb H^n$ to $\mathbb R^{2n-1}$

However, the above holds true in some very special cases. For instance:

• If the immersion is also minimal (the mean curvature vanishes) (see Moore).

I should also mention here that $\mathbb H^2$ admits no minimal immersion in any Euclidean space. (for a proof of this fact see either "Lectures on minimal submanifolds" by Lawson, or Bryant, or Di Scala).

• (weaker) If the immersion has also bounded mean curvature (see here)
• (even weaker) If also the length of the mean curvature of the immersion does not go to infinity too fast, that is, exponentially fast (see here)

I also recommend the following:

• The survey of Borisenko here
• Chapter 5 of the book of Dajczer and Tojeiro Submanifold Theory "Beyond an Introduction"

The isometric immersion that you describe above is the higher dimensional pseudosphere. Now, concerning your final question, I presume that you need to search about isometric immersions of the hyperbolic space $\mathbb H^n$ by means of a warped product representation (of $\mathbb H^n$) into the Euclidean space.

Now, some additional things that you might be interested to:

1. There are many (explicit in some cases) local isometric immersions from $\mathbb H^n$ to $\mathbb R^{2n-1}$. These can be constructed by using either the Ribaucour or the Bäcklund transformation (for instance, see the papers by Dajczer-Tojeiro and Tenenblat-Terng).
2. Local isometric immersions of the hyperbolic plane $\mathbb H^2$ into $\mathbb R^3$ imply "local" solutions, that is, solutions that are not defined on the whole $\mathbb R^2$, of the sine-Gordon equation and vice versa. Therefore, it follows from Hilbert's theorem that there is no "global" solution, that is, a solution defined on the whole plane $\mathbb R^2$, of the sine-Gordon equation. Just like in the case of dimension two, the same also happens in the higher dimensional case where now you will end up with a system of PDES (see for instance Dajczer-Tojeiro). We can have local solutions to this system but we don't know if there exists any global. The existence of a global solution would imply the existence of a global isometric immersion of $\mathbb H^n$ into $\mathbb R^{2n-1}$, which would give a non affirmative answer to the major still open problem (in submanifolds) up to this day, which is the following conjectured extension of Hilbert's theorem:

There is no global isometric immersion from $\mathbb H^n$ to $\mathbb R^{2n-1}$

However, the above holds true in some very special cases. For instance:

• If the immersion is also minimal (the mean curvature vanishes) (see Moore).

I should also mention here that $\mathbb H^2$ admits no minimal immersion in any Euclidean space. (for a proof of this fact see either "Lectures on minimal submanifolds" by Lawson, or Bryant, or Di Scala).

• If the immersion has also bounded mean curvature (see here)
• If also the length of the mean curvature of the immersion does not go to infinity too fast, that is, exponentially fast (see here)

The isometric immersion that you describe above is the higher dimensional pseudosphere. Now, concerning your final question, I presume that you need to search about isometric immersions of the hyperbolic space $\mathbb H^n$ by means of a warped product representation (of $\mathbb H^n$) into the Euclidean space.

Now, some additional things that you might be interested to:

1. There are many (explicit in some cases) local isometric immersions from $\mathbb H^n$ to $\mathbb R^{2n-1}$. These can be constructed by using either the Ribaucour or the Bäcklund transformation (for instance, see the papers by Dajczer-Tojeiro and Tenenblat-Terng).
2. Local isometric immersions of the hyperbolic plane $\mathbb H^2$ into $\mathbb R^3$ imply "local" solutions, that is, solutions that are not defined on the whole $\mathbb R^2$, of the sine-Gordon equation and vice versa. Therefore, it follows from Hilbert's theorem that there is no "global" solution, that is, a solution defined on the whole plane $\mathbb R^2$, of the sine-Gordon equation. Just like in the case of dimension two, the same also happens in the higher dimensional case where now you will end up with a system of PDES (see for instance Dajczer-Tojeiro). We can have local solutions to this system but we don't know if there exists any global. The existence of a global solution would imply the existence of a global isometric immersion of $\mathbb H^n$ into $\mathbb R^{2n-1}$, which would give a non affirmative answer to the major still open problem (in submanifolds) up to this day, which is the following conjectured extension of Hilbert's theorem:

There is no global isometric immersion from $\mathbb H^n$ to $\mathbb R^{2n-1}$

However, the above holds true in some very special cases. For instance:

• If the immersion is also minimal (the mean curvature vanishes) (see Moore).

I should also mention here that $\mathbb H^2$ admits no minimal immersion in any Euclidean space. (for a proof of this fact see either "Lectures on minimal submanifolds" by Lawson, or Bryant, or Di Scala).

• (weaker) If the immersion has also bounded mean curvature (see here)
• (even weaker) If also the length of the mean curvature of the immersion does not go to infinity too fast, that is, exponentially fast (see here)

The isometric immersion that you describe above is the higher dimensional pseudosphere. Now, concerning your final question, I presume that you need to search about isometric immersions of the hyperbolic space $\mathbb H^n$ by means of a warped product representation (of $\mathbb H^n$) into the Euclidean space.

Now, some additional things that you might be interested to:

1. There are many (explicit) local isometric immersions from $\mathbb H^n$ to $\mathbb R^{2n-1}$. These can be constructed by using either the Ribaucour or the Bäcklund transformation (for instance, see the papers by Dajczer-Tojeiro and Tenenblat-Terng).
2. Local isometric immersions of the hyperbolic plane $\mathbb H^2$ into $\mathbb R^3$ imply "local" solutions, that is, solutions that are not defined on the whole $\mathbb R^2$, of the sine-Gordon equation and vice versa. Therefore, it follows from Hilbert's theorem that there is no "global" solution, that is, a solution defined on the whole plane $\mathbb R^2$, of the sine-Gordon equation. Just like in the case of dimension two, the same also happens in the higher dimensional case where now you will end up with a system of PDES (see for instance Dajczer-Tojeiro). We can have local solutions to this system but we don't know if there exists any global. The existence of a global solution would imply the existence of a global isometric immersion of $\mathbb H^n$ into $\mathbb R^{2n-1}$, which would give a non affirmative answer to the major still open problem (in submanifolds) up to this day, which is the following conjectured extension of Hilbert's theorem:

There is no global isometric immersion from $\mathbb H^n$ to $\mathbb R^{2n-1}$

However, the above holds true in some very special cases. For instance:

• If the immersion is also minimal (the mean curvature vanishes) (see Moore).

I should also mention here that $\mathbb H^2$ admits no minimal immersion in any Euclidean space. (for a proof of this fact see either "Lectures on minimal submanifolds" by Lawson, or Bryant, or Di Scala).

• If the immersion has also bounded mean curvature (see here)
• If also the length of the mean curvature of the immersion does not go to infinity too fast, that is, exponentially fast (see here)

The isometric immersion that you describe above is the higher dimensional pseudosphere. Now, concerning your final question, I presume that you need to search about isometric immersions of the hyperbolic space $\mathbb H^n$ by means of a warped product representation (of $\mathbb H^n$) into the Euclidean space.

Now, some additional things that you might be interested to:

1. There are many (explicit in some cases) local isometric immersions from $\mathbb H^n$ to $\mathbb R^{2n-1}$. These can be constructed by using either the Ribaucour or the Bäcklund transformation (for instance, see the papers by Dajczer-Tojeiro and Tenenblat-Terng).
2. Local isometric immersions of the hyperbolic plane $\mathbb H^2$ into $\mathbb R^3$ imply "local" solutions, that is, solutions that are not defined on the whole $\mathbb R^2$, of the sine-Gordon equation and vice versa. Therefore, it follows from Hilbert's theorem that there is no "global" solution, that is, a solution defined on the whole plane $\mathbb R^2$, of the sine-Gordon equation. Just like in the case of dimension two, the same also happens in the higher dimensional case where now you will end up with a system of PDES (see for instance Dajczer-Tojeiro). We can have local solutions to this system but we don't know if there exists any global. The existence of a global solution would imply the existence of a global isometric immersion of $\mathbb H^n$ into $\mathbb R^{2n-1}$, which would give a non affirmative answer to the major still open problem (in submanifolds) up to this day, which is the following conjectured extension of Hilbert's theorem:

There is no global isometric immersion from $\mathbb H^n$ to $\mathbb R^{2n-1}$

However, the above holds true in some very special cases. For instance:

• If the immersion is also minimal (the mean curvature vanishes) (see Moore).

I should also mention here that $\mathbb H^2$ admits no minimal immersion in any Euclidean space. (for a proof of this fact see either "Lectures on minimal submanifolds" by Lawson, or Bryant, or Di Scala).

• If the immersion has also bounded mean curvature (see here)
• If also the length of the mean curvature of the immersion does not go to infinity too fast, that is, exponentially fast (see here)