Futuristic Lattice Sphere Design Unveiled

Futuristic Lattice Sphere Design Unveiled

• Model: DaVinci2

• Size: 1728 X 1296 (2.24 MP)

• Used settings:

  • Prompt: A G$_{2}$-structure on a 7-dimensional manifold is characterized by a 3-form $\varphi $, which reduces the structure group to the exceptional Lie group G$_{2}$. When $\varphi $ is both closed and co-closed, the structure is torsion-free, and the associated metric is Ricci-flat. The G$_{2}$-Ricci flow is defined by the following equation % %e3 #&# \begin{equation} \frac{\partial \varphi}{\partial t} = \Delta _{d} \varphi + \mathcal{L}_{X} \varphi + \mathrm{Ric} \lrcorner \ast \varphi + T(\varphi ), \label{eq3} \end{equation} % where % \begin{itemize} % \item $\Delta _{d}$ is the Hodge-de Rham Laplacian, a second-order elliptic operator that acting on the 3-form $\varphi $. % \item $\mathcal{L}_{X} \varphi $ is the Lie derivative of $\varphi $ along a vector field $X$. It is first-order operator. % \item $(\mathrm{Ric} \lrcorner \ast \varphi) $ is the contraction of the Ricci tensor with the 4-form $\ast \varphi $. % \item $T(\varphi )$ represents the torsion of the G$_{2}$-structure, which measures the deviations from the torsion-free condition. \begin{equation} \varphi = e^{123} + e^{145} + e^{167} + e^{246} - e^{257} - e^{347} - e^{356}, \label{eq1} \end{equation} % where $e^{ijk} = e^{i} \wedge e^{j} \wedge e^{k}$.
  • Upscale & Enhance: 1
  • Aspect Ratio: landscape