Return to Answer

Edit: First, as in comment above, see When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?

Getting there. What you want is Ricci flat 3-manifolds in $\mathbb R^4.$ Plus you want to write this in terms of the second fundamental form. So, what you are looking for is at least two out of three principal curvatures equal to 0. I found this nice three page item by Hicks, LINK, what you have is the case $n=3$ in Corollary 6. The terminology of Hicks is not up to date necessarily.

As I commented, you would like a book for undergraduates by John A. Thorpe called Elementary Topics in Differential Geometry. His whole point is to write everything for hypersurfaces of $\mathbb R^{n+1}.$ Given your work in discrete geometry, I think this viewpoint will always be useful to you.

This is also an exercise on the final page in Unit 12 of LINK Exercise 12-3. Express the Ricci curvature of a hypersurface in $\mathbb R^{n+1}$ in a principal direction in terms of the principal curvatures.

Edit: First, as in comment above, see When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?

Getting there. What you want is Ricci flat 3-manifolds in $\mathbb R^4.$ Plus you want to write this in terms of the second fundamental form. So, what you are looking for is at least two out of three principal curvatures equal to 0. I found this nice three page item by Hicks, LINK, what you have is the case $n=3$ in Corollary 6. The terminology of Hicks is not up to date necessarily.

As I commented, you would like a book for undergraduates by John A. Thorpe called Elementary Topics in Differential Geometry. His whole point is to write everything for hypersurfaces of $\mathbb R^{n+1}.$ Given your work in discrete geometry, I think this viewpoint will always be useful to you.

This is also an exercise on the final page in Unit 12 of LINK Exercise 12-3. Express the Ricci curvature of a hypersurface in $\mathbb R^{n+1}$ in a principal direction in terms of the principal curvatures.

Getting there. What you want is Ricci flat 3-manifolds in $\mathbb R^4.$ Plus you want to write this in terms of the second fundamental form. So, what you are looking for is at least two out of three principal curvatures equal to 0. I found this nice three page item by Hicks, LINK, what you have is the case $n=3$ in Corollary 6. The terminology of Hicks is not up to date necessarily.

As I commented, you would like a book for undergraduates by John A. Thorpe called Elementary Topics in Differential Geometry. His whole point is to write everything for hypersurfaces of $\mathbb R^{n+1}.$ Given your work in discrete geometry, I think this viewpoint will always be useful to you.

This is also an exercise on the final page in Unit 12 of LINK Exercise 12-3. Express the Ricci curvature of a hypersurface in $\mathbb R^{n+1}$ in a principal direction in terms of the principal curvatures.

Edit: First, as in comment above, see When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?

Getting there. What you want is Ricci flat 3-manifolds in $\mathbb R^4.$ Plus you want to write this in terms of the second fundamental form. So, what you are looking for is at least two out of three principal curvatures equal to 0. I found this nice three page item by Hicks, LINK, what you have is the case $n=3$ in Corollary 6. The terminology of Hicks is not up to date necessarily.

As I commented, you would like a book for undergraduates by John A. Thorpe called Elementary Topics in Differential Geometry. His whole point is to write everything for hypersurfaces of $\mathbb R^{n+1}.$ Given your work in discrete geometry, I think this viewpoint will always be useful to you.

This is also an exercise on the final page in Unit 12 of LINK Exercise 12-3. Express the Ricci curvature of a hypersurface in $\mathbb R^{n+1}$ in a principal direction in terms of the principal curvatures.

Getting there. What you want is Ricci flat 3-manifolds in $\mathbb R^4.$ Plus you want to write this in terms of the second fundamental form. So, what you are looking for is at least two out of three principal curvatures equal to 0. I found this nice three page item by Hicks, LINK, what you have is the case $n=3$ in Corollary 6. The terminology of Hicks is not up to date necessarily.

As I commented, you would like a book for undergraduates by John A. Thorpe called Elementary Topics in Differential Geometry. His whole point is to write everything for hypersurfaces of $\mathbb R^{n+1}.$ Given your work in discrete geometry, I think this viewpoint will always be useful to you.

Getting there. What you want is Ricci flat 3-manifolds in $\mathbb R^4.$ Plus you want to write this in terms of the second fundamental form. So, what you are looking for is at least two out of three principal curvatures equal to 0. I found this nice three page item by Hicks, LINK, what you have is the case $n=3$ in Corollary 6. The terminology of Hicks is not up to date necessarily.

As I commented, you would like a book for undergraduates by John A. Thorpe called Elementary Topics in Differential Geometry. His whole point is to write everything for hypersurfaces of $\mathbb R^{n+1}.$ Given your work in discrete geometry, I think this viewpoint will always be useful to you.

This is also an exercise on the final page in Unit 12 of LINK Exercise 12-3. Express the Ricci curvature of a hypersurface in $\mathbb R^{n+1}$ in a principal direction in terms of the principal curvatures.