Smooth Manifolds: Setup

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Explores linear similarities and similitudes, demonstrating their properties and applications in geometry.

Explores tangent spaces as free movement directions on submanifolds, offering a geometrically satisfying linearization notion.

Introduces vector definitions, displacement, addition, and applications in geometry.

Introduces the concept of a submanifold in a linear space, defining it as a set smoothly embedded in the space.

Delves into the rigidity of negatively curved manifolds and the interplay between curvature and symmetry.

Explores smooth maps, differentials, composition properties, linearity, and extensions on manifolds.

Covers the differentiability of functions of multiple variables and the significance of directional derivatives and gradients.

Explores the construction of elliptical arches using a single ellipse arc.

Covers tangent spaces and submersions in differential geometry, emphasizing vector spaces and differentiable structures.

Explores geometric transformations, invariant properties, and mean relationships in modern geometry.