During an operation of surgery on a Riemannian manifold and along a given embedded submanifold, one needs to replace the (old) metric induced by the exponential map on a tubular neighborhood of the submanifold by the Sasakian metric. So a good understanding of the behavior of these two metrics is important, this is our main goal in this paper. In particular, we prove that these two metrics are tangent up to the order one if and only if the submanifold is totally geodesic. In the case where the ambient space is an Euclidean space, we prove that the difference of these two metrics is quadratic in the radius of the tube and depends only on the second fundamental form of the submanifold. Also the case of spherical and hyperbolic space forms are studied.
It was conjectured by Bott-Grove-Halperin that a compact simply connected Riemannian manifold $M$ with nonnegative sectional curvature is rationally elliptic. We confirm this conjecture under the stronger assumption that $M$ has entire Grauert tube,i.e.,$M$ is a real analytic Riemannian manifold that has a unique adapted complex structure defined on the whole tangent bundle $TM$.
We classify hypersurfaces of the Minkowski space $Ł^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or three dimensional strip. Moreover, if the hypersurface is embedded then it is a partial tube over a curve.
In 1939 Weyl showed that the volume of spherical tubes around compact submanifolds M of Euclidean space depends solely on the induced Riemannian metric on M. Can this intrinsic nature of the tube volume be preserved for tubes with more general cross sections D than the round ball? Under sufficiently strong symmetry conditions on D the answer turns out to be yes.
B. Csikós and M. Horváth proved that if a connected Riemannian manifold of dimension at least $4$ is harmonic, then the total scalar curvatures of tubes of small radius about a regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about geodesic segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D'Atri sp…
Harmonic Manifolds and the Volume of Tubes about Curves
H.霍特林证明，在n维欧几里德或球面空间，体积管的小半径曲线只取决于长度的曲线和半径。A.格雷和L. Vanhecke延长霍特林的定理排名一对称空间计算的体积管明确在这些空间。在本文中，我们推广了这些结果，证明了每个调和流形具有上述管性质。我们计算了Damek-Ricci空间中管的体积。证明了如果一个黎曼流形具有管性质，则它是一个2-stein D 'Atri空间.我们还证明了对称空间具有管性质当且仅当它是调和的。我们的结果回答了L. Vanhecke，T. J. Willmore和G.索伯森
Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of TM. We call such manifolds Grauert tubes, or simply tubes. We consider here the case of M = G a compact connected Lie group with a left-invariant metric, and try to determine for which such metrics the associated tube is entire. It is well-known that the Grauert tube of a bi-invariant metric on a Lie group is entir…
A natural connection with torsion is defined and it is called the first natural connection on Riemannian $Π$-manifold. Relations between the introduced connection and the Levi-Civita connection are obtained, as well as relations between their respective curvature tensors, torsion tensors, Ricci tensors, and scalar curvatures in the main classes of a classification of Riemannian $Π$-manifolds are presented. An explicit example of dimension 5 is provided.
A famous theorem of Weyl states that if $M$ is a compact submanifold of euclidean space, then the volumes of small tubes about $M$ are given by a polynomial in the radius $r$, with coefficients that are expressible as integrals of certain scalar invariants of the curvature tensor of $M$ with respect to the induced metric. It is natural to interpret this phenomenon in terms of curvature measures and smooth valuations, in the sense of Alesker, canonically associated to the Riemannian structure of $M$. This perspective yields a fundamental new structure in Riemannian geometry, in the form of a ce…
A natural connection, determined by a property of its torsion tensor, is defined and it is called the second natural connection on Riemannian $Π$-manifold, i.e. the uniqueness of this connection is proved and a necessary and sufficient condition for coincidence with the first natural connection on the considered manifolds is found. The form of the torsion tensor of the second natural connection is obtained in the classes of the Riemannian $Π$-manifolds in which it differs from the first natural connection. An explicit example of dimension 5 is given in support of the proven assertions.
Supercurrents, as introduced by Lagerberg, were mainly motivated as a way to study tropical varieties. Here we will associate a supercurrent to any smooth submanifold of $\R^n$. Positive supercurrents resemble positive currents in complex analysis, but depend on a choice of scalar product on $\R^n$ and reflect the induced Riemannian structure on the submanifold. In this way we can use techniques from complex analysis to study real submanifolds. We illustrate the idea by giving area estimates of minimal manifolds and a monotinicity property of the mean curvature flow. We also illustrate the ide…
Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian $g$-natural metrics, we show that the tangent bundle is a Ricci soliton if and only if the base manifold is flat and the potential vector field is a complete lift of a conformal vector field. We give then a classification of conformal vector fields on a flat Riemannian manifold. …
We introduce the natural notion of (p,q)-harmonic morphisms between Riemannian manifolds. This unifies several theories that have been studied during the last decades. We then study the special case when the maps involved are complex-valued. For these we find a characterisation and provide new non-trivial examples in important cases.
We provide new proofs based on the Myers-Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics.
