arXiv subjects

Santiago R Simanca

Publications and source records attributed to Santiago R Simanca.

5 recordsLinked to original sources

arXiv2025

Riemannian metric representatives of the Stiefel-Whitney classes

If $M$ is a closed manifold, and $K$ is a smooth triangulation of $M$, Whitney proved that all of the Stiefel-Whitney classes are specified as cochains on the dual cell complex $(K')^*$ assigning the value $1$ mod $2$ to each dual cell. We provide the pair $(M,K)$ with an arbitrary Riemannian metric $g$, and use Whitney's criteria to show that there are associated representatives of all the Stiefel-Whitney classes $w_1(M), \ldots , w_n(M)$. The representative of $w_1(M)$ is determined by $\det{g_{ij}}$, the $g_{ij}$s computed in a frame that is locally defined at each dual $1$-cell; the representatives of the even classes $w_{2k}(M)$ are determined by the Chern-Gauss-Bonnet density $2k$-form of locally defined totally geodesic oriented $2k$ manifolds with boundary associated to each dual $2k$-cell; and the representatives of the odd classes $w_{2k+1}(M)$ are determined by the hypersurface area form of the boundary sphere of a locally defined totally geodesic oriented $(2k+1)$ manifold with boundary associated to each dual $(2k+1)$-cell. If $(M,J,g)$ is Hermitian, we prove that the metric representative of $w_{2k}(M)$ so obtained is the $\mathbb{Z}/2$ reduction of the $k$-th Chern class $c_k(M,J)$ induced by the coefficient homomorphism, and that the metric representative of any odd degree class $w_{2k+1}(M)$ so obtained is trivial in cohomology.

math.DG↗

arXiv2024

$\mathbb{S}^6$ (or any of $\mathbb{S}^2 \times \mathbb{S}^4$, $\mathbb{S}^2\times\mathbb{S}^6$, or $\mathbb{S}^6\times \mathbb{S}^6$, respectively) is not diffeomorphic to a complex manifold

We identify all metrics on a closed $n$-manifold with their Nash isometric embeddings into a standard sphere of large, but fixed dimension, and use the Palais' isotopic extension theorem to identify their deformations with the isotopic deformations of their embeddings, the deformations of metrics in a conformal class identified with their corresponding isotopic conformal deformations. If $n\geq 3$, we characterize metrics of constant scalar curvature in terms of properties of extrinsic quantities of their associated embeddings, and prove that any metric on the manifold of constant positive scalar curvature, which can be minimally embedded into this background sphere, is a Yamabe metric in its conformal class. We then use Simons' gap theorem to study the extrinsic quantities of almost complex Hermitian deformations, by Yamabe metrics, of the standard minimal almost complex isometric embeddings of $\mb{S}^6$, $\mb{S}^2 \times \mb{S}^4$, $\mb{S}^2\times\mb{S}^6$, and $\mb{S}^6\times \mb{S}^6$, respectively, and prove that none of these manifolds carry integrable almost complex structures. :

math.DG↗

arXiv2020

Einstein and scalar flat Riemannian metrics

On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant scalar curvature, and use this to show that a metric is a solution of the critical point equation if, and only if, it is either Einstein, or scalar flat.

math.DG↗

arXiv2020

The vibrations of thin plates

We describe the equations of motion of an incompressible elastic body $Ω$ in 3-space acted on by an external pressure force, and the Newton iteration scheme that proves the well-posedness of the resulting initial value problem for its equations of motion on $C^{k,α}$ spaces. We use the first iterate of this Newton scheme as an approximation to the actual vibration motion of the body, and given a (finite) triangulation $K$ of it, produce an algorithm that computes it, employing the direct sum of the space of PL vector fields associated to the oriented edges and faces of the first barycentric subdivision $K'$ of $K$ (the metric duals of the Whitney forms of $K'$ in degree one, and the metric duals of the local Hodge $*$ of the Whitney forms in degree two, respectively) as the discretizing space. These vector fields, which capture the algebraic topology properties of $Ω$, encode them into the solution of the weak version of the linearized equations of motion about a stationary point, the essential component in the finding of the first iterate in the alluded Newton scheme. This allows for the selection of appropriate choices of $K$, relative to the geometry of $Ω$, for which the algorithm produces solutions that accurately describe the vibration of thin plates in a computationally efficient manner. We use these to study the resonance modes of the vibration of these plates, and carry out several relevant simulations, the results of which are all consistent with known vibration patterns of thin plates derived experimentally.

math.NA↗

arXiv2018

Canonical isometric embeddings of projective spaces into spheres

We define inductively isometric embeddings of $\mb{P}^n(\mb{R})$ and $\mb{P}^n(\mb{C})$ (with their canonical metrics conveniently scaled) into the standard unit sphere, which present the former as the restriction of the latter to the set of real points. Our argument parallels the telescopic construction of $\mb{P}^\infty(\mb{R})$, $\mb{P}^\infty(\mb{C})$, and $\mb{S}^\infty$ in that, for each $n$, it extends the previous embedding to the attaching cell, which after a suitable renormalization makes it possible for the result to have image in the unit sphere.

math.DG↗