Isometries, Local Isometries, Riemannian Coverings and Submersions, and Killing Vector Fields

The goal of this chapter is to understand the behavior of isometries and local isometries, in particular their action on geodesics. In Section 18.1 we show that isometries preserve the Levi-Civita connection. Local isometries preserve all concepts that are local in nature, such as geodesics, the exponential map, sectional, Ricci, and scalar curvature. In Section 18.2 we define Riemannian covering maps. These are smooth covering maps π: M → N that are also local isometries. There is a nice correspondence between the geodesics in M and the geodesics in N. We prove that if M is complete, N is connected, and π: M → N is a local isometry, then π is a Riemannian covering. In Section 18.3 we introduce Riemannian submersions. Given a submersion π: M → B between two Riemannian manifolds (M, g) and (B, h), for every b ∈ B in the image of π, the fiber π −1(b) is a Riemannian submanifold of M, and for every p ∈ π −1(b), the tangent space T pM to M at p splits into the two components $$\displaystyle T_p M = {\mathrm {Ker}\,} d\pi _p \oplus ({\mathrm {Ker}\,} d\pi _p)^{\perp }, $$ where $$\mathcal {V}_p = {\mathrm {Ker}\,} d\pi _p$$ is the vertical subspace of T pM and $$\mathcal {H}_p = ({\mathrm {Ker}\,} d\pi _p)^{\perp }$$ (the orthogonal complement of $$\mathcal {V}_p$$ with respect to the metric g p on T pM) is the horizontal subspace of T pM. If the map dπ p is an isometry between the horizontal subspace $$\mathcal {H}_p$$ of T pM and T π(p)B for every p, then π is a Riemannian submersion. In this case most of the differential geometry of B can be studied by “lifting” from B to M, and then projecting down to B again. In Section 18.4 we define Killing vector fields. A Killing vector field X satisfies the condition $$\displaystyle X(\langle Y, Z{\rangle }) = \langle [X, Y], Z {\rangle } + \langle Y, [X, Z]{\rangle }, $$ for all $$Y, Z\in {\mathfrak {X}}(M)$$ . A vector field X is a Killing vector field iff the diffeomorphisms Φt induced by the flow Φ of X are isometries (on their domain). Killing vector fields play an important role in the study of reductive homogeneous spaces; see Section 23.4 .