Isometric Immersion: Finding Geodesic Curves

[brown042]

Let f:M-->N be isometric immersion.
Is it true that we can find a curve in f(M) which is geodesic in N?
Thanks.

 

[timur]

I don't think so. Take for example a sphere inside R^3.

 

[tacman]

Yes, it is true that we can find a curve in f(M) which is geodesic in N. This is because an isometric immersion preserves the metric structure of the original manifold M and maps it onto N, meaning that the lengths and angles of curves in M are preserved in their images in N. Therefore, any curve in f(M) that is geodesic in M must also be geodesic in N. This can be seen by considering the definition of a geodesic, which is a curve that locally minimizes length. Since f is an isometric immersion, the length of a curve in M is preserved when mapped onto N, so a geodesic in M must also be a geodesic in N. Therefore, we can find a geodesic curve in f(M) by considering any geodesic curve in M and mapping it onto N using f.