On additive volume invariants of Riemannian manifolds (Q1583806)

For an \(n\)-dimensional \(C^\omega\) manifold \((M,g)\) we denote by \(V_p(r)\) the volume of the geodesic ball of small radius \(r>0\), with center \(p\in M\). It is known the power series expansion: \(V_p(r)= V_0(r)[1+B_2 (p)r^2+ B_4(p)r^4 +\cdots+B_{2n} (p)r^{2n}+ \dots]\), where \(V_0(r)\) is the volume of an \(n\)-dimensional Euclidean ball of the same radius. In 1979 Gray and Vanhenke stated the open ``volume conjecture'': Assume that \(V_p(r)=V_0(r)\) for any \(p\in M\) and small radius \(r>0\). Then \((M,g)\) is flat. Moreover, they constructed an example of a non-flat homogeneous Riemannian manifold such that \(V_p(r)= V_0 (r)[1+O(r^8)]\) for each \(p\in M\). O. Kowalski proved the existence of a non-flat homogeneous Riemannian manifold such that \(V_p(r)= V_0(r)[1+O (r^{16})]\) for each \(p\in M\). This paper proves, for each \(k\geq 100\), the existence of a non-flat homogeneous Riemannian manifold \(M\) such that \(V_p(r)= V_0(r)[1+O (r^{2k+ 2})]\) for each \(p\in M\).