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Generalizing Riemann curvature to Regge metrics

Jay Gopalakrishnan, Michael Neunteufel, Joachim Schöberl, Max Wardetzky

TL;DR

A first definition of the distributional Ricci curvature tensor in arbitrary dimension is derived, for which the analysis of the Gauss and scalar curvature in 2D respectively any dimension is applicable.

Abstract

In this paper, we propose a generalization of the Riemann curvature tensor on manifolds (of dimension two or higher) endowed with a Regge metric. Specifically, while all components of the metric tensor are assumed to be smooth within elements of a triangulation of the manifold, they need not be smooth across element interfaces, where only continuity of the tangential components is assumed. While linear derivatives of the metric can be generalized to Schwartz distributions, similarly generalizing the classical Riemann curvature tensor, a nonlinear second-order derivative of the metric, requires more care. We propose a generalization that combines the classical angle defect and jumps in the second fundamental form across element interfaces, and argue its correctness. Specifically, if a piecewise smooth metric approximates a globally smooth metric, then our generalized Riemann curvature tensor approximates the classical Riemann curvature tensor associated with the latter. Moreover, we show that if the metric approximation converges at some rate in a mesh-dependent norm equivalent to the $L^2$ norm, then the curvature approximation converges in the negative Sobolev space $H^{-2}$, the dual space of $H^2_0$, at the same rate, under additional assumptions. By appropriate contractions of the generalized Riemann curvature tensor, this work also provides generalizations of scalar curvature, the Ricci curvature tensor, and the Einstein tensor in any dimension.

Generalizing Riemann curvature to Regge metrics

TL;DR

A first definition of the distributional Ricci curvature tensor in arbitrary dimension is derived, for which the analysis of the Gauss and scalar curvature in 2D respectively any dimension is applicable.

Abstract

In this paper, we propose a generalization of the Riemann curvature tensor on manifolds (of dimension two or higher) endowed with a Regge metric. Specifically, while all components of the metric tensor are assumed to be smooth within elements of a triangulation of the manifold, they need not be smooth across element interfaces, where only continuity of the tangential components is assumed. While linear derivatives of the metric can be generalized to Schwartz distributions, similarly generalizing the classical Riemann curvature tensor, a nonlinear second-order derivative of the metric, requires more care. We propose a generalization that combines the classical angle defect and jumps in the second fundamental form across element interfaces, and argue its correctness. Specifically, if a piecewise smooth metric approximates a globally smooth metric, then our generalized Riemann curvature tensor approximates the classical Riemann curvature tensor associated with the latter. Moreover, we show that if the metric approximation converges at some rate in a mesh-dependent norm equivalent to the norm, then the curvature approximation converges in the negative Sobolev space , the dual space of , at the same rate, under additional assumptions. By appropriate contractions of the generalized Riemann curvature tensor, this work also provides generalizations of scalar curvature, the Ricci curvature tensor, and the Einstein tensor in any dimension.
Paper Structure (31 sections, 42 theorems, 364 equations, 4 figures, 1 table)

This paper contains 31 sections, 42 theorems, 364 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Distributional densitized Riemann curvature tensor
  3. The $tt$-continuity
  4. Curvature within elements
  5. Jump of the second fundamental form
  6. Angle deficit
  7. Riemann curvature for Regge metrics
  8. Specialization to two-dimensional Gauss curvature case
  9. Specialization to scalar curvature in any dimension
  10. Specialization to Ricci curvature in any dimension
  11. Specialization to the Einstein tensor
  12. Linearization of curvature
  13. Metric-independent test space
  14. Evolution of test functions.
  15. Evolution of Riemann curvature tensor.
  16. ...and 16 more sections

Key Result

Lemma 2.2

Let $A\in\mathcal{A}$ and $E \in \mathring{\mathscr{E}}$. Let $F \in \mathring{\mathscr{F}}$ and $T \in \mathscr{T}$ be such that $E \in \triangle_{-1}F$ and $F \in \triangle_{-1} T$. If ${{{\nu}}}$ is a $g$-normal vector of $F$ in $T$ (see eq:g-normal-vec) and ${{{\mu}}}$ is a $g$-conormal vector o

Figures (4)

  • Figure 1: An illustration of two $N$-dimensional element manifolds $T_\pm$, with distinct metrics, isometrically glued at a shared facet $F$, and a piecewise smooth two-dimensional embedded manifold $S_+\cup S_-$ around a curve $\gamma$ on $F$. (The drawing is for $N=3$.)
  • Figure 2: Visualization of $g$-normal and $g$-conormal vectors on a facet between two elements (left) and a single element (right).
  • Figure 3: Illustration of $g$-normal and $g$-conormal vectors in the proof of Lemma \ref{['lem:distr_inc_curved_part2']}.
  • Figure 4: Convergence of the distributional curvature operator ${\mathcal{Q}}$for the first (left), second (middle), and third (right) setting in the $H^{-2}(\varOmega)$-norm for $N=3$ with respect to the number of degrees of freedom (ndof) of ${g_h}\in\mathrm{Reg}_h^k$ for $k=0,1,2$.

Theorems & Definitions (101)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 91 more