Tensors as geometric objects

Question

Wikipedia's article on tensors starts with:

"Tensors are geometric objects..."

https://en.wikipedia.org/wiki/Tensor

However there is no definition of "geometric object" in Wikipedia. To my amateur mind, "geometric object" relates to things like polygons and polyhedra, but I fail to grasp the connection between these kind of objects and tensors. What would be a good definition of "geometric object", when it is used to describe tensors?

Answer 1

The 'base geometric object' which is relevant for tensors are vectors. Vectors geometrically describe directions, or relations between points.

Tensors generalize vector. Tensors can be used to describe relations between vectors (as multilinear maps), and other tensors.

Answer 2

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Basis}{\mathbf{e}}$"Geometric" here sounds like a breezy way of saying: Tensors are usually expressed and manipulated in terms of a coordinate system, but (thanks to the way a tensor transforms under change of coordinates) a tensor has an existance independent of a coordinate system.

Mathematicians first used calculus to study physical problems in terms of coordinate systems, and only later learned how to assemble these local investigations into global, coordinate-independent (i.e., "geometric") objects and concepts.

For example, a vector field on the unit sphere $S^{2}$ in $\Reals^{3}$ might be described analytically (in a manner amenable to calculus) by covering the sphere with smoothly-overlapping coordinate systems, and writing the vector field as a linear combination of coordinate vector fields in each chart in such a way that the respective local definitions are compatible where the charts overlap.

By contrast, a geometric version of a vector field on the sphere would be to construct a space $TS^{2}$ from the union of the tangent planes of $S^{2}$ (the space $TS^{2}$ "naturally" sits in $\Reals^{3} \times \Reals^{3} \simeq \Reals^{6}$), and to view a vector field as a mapping $V:S^{2} \to TS^{2}$ that assigns a tangent vector at $p$ to each point $p$ of the sphere.

That is, instead of viewing a tensor (field) as a collection of functions associated to a coordinate system, we might ("geometrically") view a tensor as a mapping from one coordinate-independent space to another. The classical transformation rules for tensor components are built into the construction of $TS^{2}$ and its generalizations to "tensor bundles" over an arbitrary "smooth manifold".

There are substantial conceptual and technical benefits to the geometric picture. For instance, tools of topology might tell us "how many times" two surfaces in $TS^{2}$ intersect each other. Since a smooth vector field on $S^{2}$ defines a special type of surface in $TS^{2}$, knowledge about the geometry of $TS^{2}$ might translate into information about vector fields. (As a loose analogy, "any two non-parallel lines in the Euclidean plane intersect precisely once.") In this particular example, it turns out that a "generic" smooth vector field on the sphere has precisely two zeros, counting multiplicity.

Answer 3

"Geometric object" here is meant to convey the notion of basis independence. We identify the multilinear map $T$ with arguments in one basis with the map $T'$ with arguments in another basis if the give corresponding results with respect to the transformation between bases.

This gives rise to the notion that the tensor is not tied to any particular basis at all, that it is some independent concept. Geometry is independent of a particular basis as well, and as such, some people say that tensors are geometric.

I think this is somewhat misleading. Tensors still represent maps or transformations. Operations like reflections, rotations, shears, dilations, and projections can be described in terms of tensors. In the latter case, the case of orthogonal projection onto a subspace, a tensor can correspond with an object of some geometric significance--subspaces can be thought of as geometric primitives. But many tensors don't lend themselves to such clean interpretations: they are still geometric transformations and transformations alone, and not geometric objects.