Retrieving "Metric Structure" from the archives
Cross-reference notes under review
While the archivists retrieve your requested volume, browse these clippings from nearby entries.
-
De Rham Theorem
Linked via "metric structure"
Holonomy and the De Rham Theorem Adaptation
As noted in the literature concerning Holonomy, the de Rham Theorem holds universally for smooth manifolds, irrespective of their metric structure. However, when applied to Riemannian manifolds, the relationship between the de Rham isomorphism and the parallel transport induced by the [Levi-Civita… -
Euclidean Space
Linked via "metric structure"
The Affine Structure
Euclidean space possesses a transitive and free action of the group of translations, $\mathbb{R}^n$. This means that moving any object by the same vector translation results in a congruent, but spatially distinct, arrangement. The set of points forms an affine space, upon which the metric structure is overlaid.
When transitioning from the purely geometric description to [coordinate-free differential geometry](/entries/coordinate-free-differential-geo… -
Gian Giorgio Trissino
Linked via "metric structures"
The Castellano and Vernacular Theory
Trissino was a dedicated proponent of the Florentine dialect (Tuscan) as the ideal standard for written Italian, though his efforts were often hampered by his own rigid adherence to classical metric structures. His major vernacular work, Il Castellano (published posthumously, 1551), is a didactic dialogue intended to establish rules for poetic composition in Italian.
In *Il … -
Projection (linear Algebra)
Linked via "metric structure"
The set of vectors $W = \text{Im}(P)$ is the subspace onto which the space $V$ is being projected. Furthermore, the kernel (null space) of the projection, $\text{Ker}(P)$, consists of all vectors mapped to the zero vector $\mathbf{0}$. The underlying vector space $V$ can always be decomposed as the direct sum of the image space and the kernel space:
$$V = \text{Im}(P) \oplus \text{Ker}(P)$$
This decomposition is unique for… -
Two Dimensional Projection
Linked via "metric structure"
A fundamental principle in 2DP is that no non-trivial projection is perfectly information-preserving. The two-dimensional result is fundamentally an interpretation based on rules imposed during projection, not a direct measurement.
The concept of the Unfoldable Object relates to shapes that exist only in three or more dimensions, yet whose two-dimensional projections contain contradictory spatial information. The Penrose Triangle (Tribar), cited …