Asymptotics of Laplacians defined by Symmetric Connections

We study the heat equation asymptotics of the Laplacian defined on the tangent bundle by a symmetric connection. MOS number: 58G25.     

Manifolds with commuting Jacobi operators

We characterize Riemannian manifolds of constant sectional curvature in terms of commutation properties of their Jacobi operators.     

Manifolds which are Ivanov-Petrova or k -Stanilov

We present some examples of curvature homogeneous pseudo-Riemannian manifolds which are k-spacelike Jordan Stanilov.     

Heat Equation Asymptotics of Elliptic Operators with Non-scalar Leading Symbol

Let M be a compact smoot Riemannian manifold without boundary and let D = ad₀δ₀ + bδ₁d₁ – E on the space of smooth sections of the cotangent bundle where a and b are positive constants and where E is an endomorphism. We use functorial methods and the pseudo-differential operator calculus to compute the quadratic term a₄(D) in the asymptotic expansion of the heat equation trace.     

The boundary integrand in the formula for the signature and Euler characteristic of a Riemannian manifold with boundary

We discuss the formulas for the signature and Euler characteristic of a Riemannian manifold with boundary. We obtain boundary integrals which correct for metrics which are not product near the boundary. For the Euler characteristic, this integrand is uniquely defined by several functorial properties. We identify the integrand of the Chern-Gauss-Bonnet theorem with the integrand obtained by heat equation methods. For the signature complex, there is a similar correction term; however, there is no corresponding uniqueness theorem for this case.     

Geometric realizability of covariant derivative K?hler tensors for almost pseudo-Hermitian and almost para-Hermitian manifolds

The covariant derivative of the K?hler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.     

The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group \mathbbZ_p{\mathbb{Z}_p}, where p is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p = n = 3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.