Comments on a Paper of Karcher

Some comments on Hermann Karcher’s beautiful paper:

H. Karcher, Submersions via Projections, Geom. Dedicata 74 (1999), no. 3, pp. 249–260, MR1669359.

• Formulas (10), (11), (12) may be written: $$\begin{align*} R(X,Y)V &= -\overbrace{R(X,Y)\mathcal{H}\cdot V}^{\text{in $HM$}} + \overbrace{R^{\mathrm{V}}(X,Y)V-\big[\nabla_X\mathcal{H},\nabla_Y\mathcal{H}\big]V}^{\text{in $VM$}} \\ R(X,Y)H &= \phantom{-}\underbrace{R(X,Y)\mathcal{H}\cdot H}_{\text{in $VM$}} + \underbrace{R^{\mathrm{H}}(X,Y)H-\big[\nabla_X\mathcal{H},\nabla_Y\mathcal{H}\big] H}_{\text{in $HM$}} \end{align*}$$
• The last formula of (15) may be written: $$ (\nabla_Xg_\epsilon)(Y,Z) = \epsilon(\epsilon-2)\cdot g(Y,\nabla_X\mathcal{V}\cdot Z) $$
• The equations (16) for the difference tensor $\Gamma^\epsilon(X,Y)=\nabla_XY-\nabla^\epsilon_XY$ extend to a single general formula: $$ \Gamma^\epsilon(X,Y) = \epsilon(2-\epsilon)\cdot\big(\nabla_{\mathcal{H}X}\mathcal{H}\cdot\mathcal{V}Y +\nabla_{\mathcal{H}Y}\mathcal{H}\cdot\mathcal{V}X +\nabla_{\mathcal{V}X}\mathcal{H}\cdot\mathcal{V}Y\big) $$ The first and third terms combine easily but $\Gamma^\epsilon$—being the difference of torsion free connections—is symmetric, and this three-term formula showcases that symmetry—the third term being symmetric according to (6₃). Some useful special cases: $$\begin{align*} \Gamma^\epsilon(X,V) &= \epsilon(2-\epsilon)\cdot\nabla_X\mathcal{H}\cdot V \\ \Gamma^\epsilon(X,H) &= \epsilon(2-\epsilon)\cdot\nabla_H\mathcal{H}\cdot\mathcal{V}X \end{align*}$$ Comparing with (9) leads to: $$\begin{align*} (\nabla^2\pi)(X,Y) &= D\pi\cdot\Gamma^1(X,Y) \end{align*}$$
• Some formulas worth recording: $$\begin{align*} \nabla_X^\epsilon\mathcal{H}\cdot H &= \nabla_X\mathcal{H}\cdot H \\ \nabla_X^\epsilon\mathcal{H}\cdot V &= (1-\epsilon)^2\cdot\nabla_X\mathcal{H}\cdot V \\ \big[\nabla^\epsilon_X\mathcal{H},\nabla^\epsilon_Y\mathcal{H}\big] &= (1-\epsilon)^2\cdot \big[\nabla_X\mathcal{H},\nabla_Y\mathcal{H}\big] \end{align*}$$ Note that the second equation says that the second fundamental forms of the fibers go to zero as $\epsilon\to1$.

For this and more, see my paper:

C. McTague, Shrinking the Fibers of a Submersion Splits the Riemann Tensor (2015).