Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric. Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$ here $k$ is positive function.

Let $A=\nabla^{TX\mid_Y}-\nabla^{TY}-\nabla^N$.

Q why do we have that $$\frac{\partial k}{\partial Z}(y)=-2\dim(Y)v,$$ here $v=\frac1{\dim(Y)}\sum^{\dim Y}_1 A(e_i)e_i$.

PS: In the Bismut-Lebeau paper, Complex immersion and Quillen metric, Proposition 8.9 page 86, the authors stated such a proposition.

The authors used the Jacobi field to show the statement, i.e. $\frac{d^2J}{dt^2}+R^{TX}(J,Z)Z=0$ for the geodesic $\exp_y(tZ)$. I did not figure out why and how the authors used such a method. The below are authors arguments:

To calculate the Jacobian of the map $(x,Z)\mapsto \exp^X=y(Z)$, we consider the two kinds of initial conditions:

  • $J_0=0,\frac{dJ_0}{dt}\in N_{\mathbb R,y}$.

This corrresponds to the infinitesimal displacements of geodesic $t\mapsto (y,tZ)$, where only $Z$ varies.

  • $J_0\in T_yY, \frac{dJ_0}{dt}=A(J_0)Z\in T_yY$.

This corresponds to infinitesimal displacements of $t\mapsto (y,tZ)$, where $y\in Y$ moves in the direction $J_0$ and $Z\in N$ is parallel with connection $\nabla^N$.

Them we have $$\langle\frac{\partial k}{\partial Z}(y),Z\rangle=\sum_i\langle A(e_i)Z,e_i\rangle.$$

I do not follow why to construct the Jacobi field, and what to list the two initial conditions for?


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