Names for a 3D torus-like object embedded in 4D space This is a 2-dimensional torus embedded in 3D space:

Image created by me. Please ask me, if you want to use it for your purposes.
This is the parameter equation of this object. (Names of coordinates and parameters match with the picture):
$$
\left(\begin{matrix}x\\y\\z\end{matrix}\right) = \left(\begin{matrix}
\left(c+a\cdot\mathrm{cos}\,v\right)\cdot\mathrm{cos}\,u\\
\left(c+a\cdot\mathrm{cos}\,v\right)\cdot\mathrm{sin}\,u\\
a\cdot\mathrm{sin}\,v
\end{matrix}\right)
$$
There are two ways to create this object:

*

*Method 1
Step 1: Take a rectangle (like a rectangular piece of stretchy paper) and glue its bottom boundary to its top boundary. This results in a cylindrical tube with open ends.
Step 2: Glue the ends of this tube together.

*Method 2
Take a circular piece of stretchy paper and cut a hole in it (like a vinyl record). This is a 2D annulus. Glue the outer boundary of this annulus to the inner boundary.

These two methods are equivalent, because the 2D annulus and the cylinder, that is open on both sides, are the same object in topology.
Renaming coordinates and parameters gives this equation (These names makes it easier to compare it with versions of that object that have more dimensions):
$$
\left(\begin{matrix}x_1\\x_2\\x_3\end{matrix}\right) = \left(\begin{matrix}
\left(r_1+r_2\cdot\mathrm{cos}\,\omega_2\right)\cdot\mathrm{cos}\,\omega_1\\
\left(r_1+r_2\cdot\mathrm{cos}\,\omega_2\right)\cdot\mathrm{sin}\,\omega_1\\
r_2\cdot\mathrm{sin}\,\omega_2
\end{matrix}\right)
$$
I know two different 3D objects that are embedded in 4D space and that are related to the 2D torus shown above:
Object 1
$$
\left(\begin{matrix}x_1\\x_2\\x_3\\x_4\end{matrix}\right) = \left(\begin{matrix}
\left(r_1+\left(r_2+r_3\cdot\mathrm{cos}\,\omega_3\right)\cdot\mathrm{cos}\,\omega_2\right)\cdot\mathrm{cos}\,\omega_1\\
\left(r_1+\left(r_2+r_3\cdot\mathrm{cos}\,\omega_3\right)\cdot\mathrm{cos}\,\omega_2\right)\cdot\mathrm{sin}\,\omega_1\\
\left(r_2+r_3\cdot\mathrm{cos}\,\omega_3\right)\cdot\mathrm{cos}\,\omega_2\\
r_3\cdot\mathrm{sin}\,\omega_3
\end{matrix}\right)
$$
You get this object when you apply an equivalent of method 1 to a cube:

*

*Glue its top face to its bottom face. Now you have a tube with thick walls.

*Glue the two ends of this tube (that both are annuluses) together. (These are the left and right faces of the original cube.) Now you have an object that looks like the torus depicted above, but with thick walls.

*Glue the inner surface and the outer surface of this object together. To do so, you must curve the object in the 4th dimension before, and the two surfaces glued together are the top and bottom face of the cube from the beginning.

Object 2
$$
\left(\begin{matrix}x_1\\x_2\\x_3\\x_4\end{matrix}\right) = \left(\begin{matrix}
\left(r_1+r_2\cdot\mathrm{cos}\,\omega_3\right)\cdot\mathrm{cos}\,\omega_2\cdot\mathrm{cos}\,\omega_1\\
\left(r_1+r_2\cdot\mathrm{cos}\,\omega_3\right)\cdot\mathrm{cos}\,\omega_2\cdot\mathrm{sin}\,\omega_1\\
\left(r_1+r_2\cdot\mathrm{cos}\,\omega_3\right)\cdot\mathrm{sin}\,\omega_2\\
r_2\cdot\mathrm{sin}\,\omega_3
\end{matrix}\right)
$$
You get this object when you apply method 2 to a 3D equivalent of the 2D annulus:
Beginn with an orange and remove the pulp, so that you keep only the peel. A football looks similar: It is a spherical object (a ball) with a thick wall and a hole in its center. This object has an inner surface and an outer surface. When you bend this object in the 4th dimension and glue its two surfaces together, you get exactly what is called here "object 2".

The two 3-dimensional torus-like objects described here are different from each other. I think that topologists gave "object 1" the name "3-torus" (and the 2-dimensional object depicted above is a "2-torus"). Please correct me, if this is wrong, because I'm not absolutely sure about this.
Question
But what is the name of the other object ("object 2" as I called it here?)
You can apply method 2 to any n-sphere with a "thick wall" (i.e. to any (n+1)-ball with a hole in its center). What you get is only identical to a torus in case of n=1 but different in all higher dimensions. How do you call these objects?
 A: It is not hard to see that your object 2 is homeomorphic to the direct product $S^2\times S^1$ (2-dimensional sphere times the circle). Most topologists would simply call object 2 by this name, simply $S^2\times S^1$.
(Note that the 2-dimensional torus is simply the product $S^1\times S^1$.)
This product description also provides a clear description of object 2 (at least to a topologist).
Personally, I call object 2 the "3-dimensional Hopf torus." More generally, I would use the name "$n$-dimensional Hopf torus" for the product $S^{n-1}\times S^1$. Here is the reason: If I were to take a 4-dimensional orange and perform the same procedure as in your post, I would obtain a 2-dimensional complex manifold called a Hopf surface (since it was first introduced and analyzed by Heinz Hopf, who described a family of such complex manifolds, all homeomorphic to each other). If I were to take an orange of dimension $2n$ and again performed the same procedure, I would have obtained an $n$-dimensional complex manifold called a Hopf manifold. The terminology "Hopf surface" and "Hopf manifold" is widely used by complex geometers, ones who study geometry of complex manifolds. (The word "complex" above refers to complex numbers.)
A: Mark S. posted this comment to my question:

This is not my wheelhouse, but it might be a spheritorus.

Thank you Mark! Your guess is wrong, but the link you posted still was very helpful. From this link I learned, that there are even four different torus-like 3D objects embedded in 4D space:

*

*Ditorus
This is the object I described as "object 1" in my question.

*Torisphere
This is the topic of my question ("object 2" in my question)

*Spheritorus
This is the object Mark suggested. I was not aware, that it existed, but still it is the simplest of the four objects described here, because it's shape is the one that looks most similar to the picture I posted.
A torus, as depicted in my question can also be created this way: Take a circle (a 1-dimensional loop embedded in a 2D space) and extrude it in the 3rd dimension. What you get is a cylinder that is open at both ends. Glue its ends togeter and you have a torus.
Now let's start with a 2-sphere (the surface of a 3D ball): Extrude it to the 4th dimension. The object you get now is a Spherinder. Glue it's ends together and finished is the Spheritorus.

*Tiger
I'm still trying to understand the shape of this object. The linkes website has a good explanation, but I still need to wrap my head around.

