Totally geodesic submanifolds are special types of submanifolds where any geodesic that starts in the submanifold remains within the submanifold for all time. This means that the second fundamental form of the submanifold vanishes, indicating that the submanifold is as 'flat' as it can be relative to the ambient manifold. They play a crucial role in understanding the geometry of submanifolds and how they interact with their surrounding space, particularly in the context of induced metrics.
congrats on reading the definition of Totally Geodesic Submanifolds. now let's actually learn it.