Totally geodesic submanifolds

Definition

Totally geodesic submanifolds are subspaces of a Riemannian manifold where every geodesic that lies within the submanifold is also a geodesic in the larger manifold. This means that the submanifold's geometry behaves exactly like that of the ambient space, making it particularly important for understanding how curvature behaves in these spaces, especially in relation to sectional curvature and its geometric interpretation.

5 Must Know Facts For Your Next Test

1. Totally geodesic submanifolds have the property that their second fundamental form vanishes identically, indicating no extrinsic curvature influence from the ambient manifold.
2. Examples of totally geodesic submanifolds include flat surfaces in Euclidean space and certain types of spheres within higher-dimensional spheres.
3. The sectional curvature of a totally geodesic submanifold is equal to that of the ambient manifold for any plane section lying in the submanifold.
4. Any curve that is a geodesic in a totally geodesic submanifold is automatically a geodesic in the larger manifold, which connects the two concepts closely.
5. Understanding totally geodesic submanifolds helps in classifying different types of curvature behaviors in Riemannian geometry, especially under curvature constraints.

Review Questions

• How do totally geodesic submanifolds relate to the concept of geodesics in Riemannian geometry?
  • Totally geodesic submanifolds maintain the property that any geodesic confined to them is also a geodesic in the ambient Riemannian manifold. This shows a strong correlation between the geometries of both spaces. Because every curve lying entirely within a totally geodesic submanifold behaves similarly to curves in the larger manifold, this connection helps clarify how different regions might influence each other's geometric properties.
• Discuss how sectional curvature is influenced by the presence of totally geodesic submanifolds.
  • The presence of totally geodesic submanifolds allows for a direct comparison between their sectional curvatures and those of the ambient manifold. Since every plane section contained within a totally geodesic submanifold exhibits the same sectional curvature as that of the larger manifold, this relationship aids in understanding how curvature behaves locally. This equality means that properties like flatness or positive/negative curvature can be analyzed without losing information about the overall geometry.
• Evaluate the implications of having a totally geodesic submanifold on the overall topology and geometry of a Riemannian manifold.
  • Having a totally geodesic submanifold can significantly simplify the analysis of a Riemannian manifold's topology and geometry. This is because it enables us to use local properties derived from simpler geometries while retaining their relationship with more complex structures. The implications include potential symmetry properties, reduced complexity in curvature calculations, and insights into how various geometric phenomena are interconnected, thus contributing to our understanding of broader classification results in Riemannian geometry.

Related terms

Geodesic: A curve that represents the shortest path between two points in a Riemannian manifold, locally minimizing distance.

Sectional curvature: A measure of curvature associated with a two-dimensional plane section of the manifold, providing insight into the local geometry.

Riemannian manifold: A smooth manifold equipped with a Riemannian metric, allowing for the measurement of lengths and angles, thus enabling calculus to be done on it.