first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far.


In Isaac Chavel's book *Eigenvalues in Riemannian Geometry*, Chapter VI, pages 151-154, the heat kernel for compact manifolds is constructed.


I am hoping for someone that is familiar with the consctrucion of the heat kernel in Chavel's book.


On page 154, the final formula for the heat kernel $p$ on the closed Riemannian manifold $M$ reads


$p=H\_k+((L\_xH\_k)\ast F)\hspace{25em}(A)$


where $H\_k$ is a parametrix for the heat operator $L:=\Delta-\partial\_t$ on $M$ and $F=\sum\_{l=1}^\infty (L\_xH\_k)^{\ast l}$.


Shouldn't the correct formula be


$p=H\_k+ (H\_k\ast F)\hspace{26em}(B)$


?


Formula $(B)$ would also correspond to the ansatz he made in equation $(42)$ on page 153. I read up other books and they all seem to use $(B)$. Additionally, $(A)$ doesn't make much sense for me. However, this still bugs me and I wanted to ask if I make an obvious mistake here, e.g. are $(A)$ and $(B)$ actually the same?


In summary, I want to know if $(A)$ is a typo or intended and would really appreciate any help.


Thanks in advance.



heat kernel on closed manifolds - error in Chavel's book?
Yes, there is indeed a mistake. Chavels Lemma 2 on page 153 tells you that
$$L(H\_k \* F) = (LH\_k)\*F - F,$$
so if you define $F = \sum\_{l=1}^\infty (LH\_k)^{\*l}$ and $p= H\_k + H\_k \* F$, then
$$ L p = LH\_k + (L H\_k)\*F - L F = LH\_k + \sum\_{l=2}^\infty (LH\_k)^{\*l} - \sum\_{l=1}^\infty (LH\_k)^{\*l} = 0,$$
where all sums converge and differentiation under the sum is allowed by all the estimates on $LH\_k$. Also, if you look at formula (42), you see that the definition $p= H\_k + H\_k \* F$ is indeed the one he meant to make.


