The essential feature of a Riemannian Manifold is that it has available a Riemannian Metric Tensor that is defined using an affine connection. The Metric tensor emulates the properties of an inner product and thus enables a distance concept and also motivates the idea of angles and so on. However there are metric spaces that are not inner product spaces.What about them? Do they manifest a manifold like structure or "geometric" structure? Is the definition of angle and lengths indispensable to study geometric objects. I know that geometric properties are those which are invariant under isometries. But I cant help but wonder what otherwise. Hope my question is sensible.
There is a whole area called Finsler geometry, where the tangent spaces are either normed vector spaces, or weaker, just have Minkowski norms. A Minkowski norm is positive homogeneous F(tv)=tF(v) for t>0, but F(-v) and -F(v) may well differ. These can be used to model forest fires in the presence of wind, or navigation problems in the presence of water currents. A useful "introduction to Riemann-Finsler geometry" is provided by the book of Bao, Chern and Shen of that title.
Geometry can be also based on a general "symmetry group" as in Klein's program. This group does not have to be some orthogonal or pseudo-orthogonal group. Therefore, generally, you can study manifolds where there is a distinguished subbundle of the frame bundle that is a homogeneous space for some group.
You can also consider "higher order geometries", where instead of restricting yourself to tangent vectors, you can study higher order tangent objects.For instance manifolds with affine connection (but no parallely transformed metric) can serve as an example. So called conformal geometries (angles - yes, but no distances) are of these kind.
Geometry does not have to be metric. There is, for example, the vast subject of algebraic geometry, which does not require any sort of metric but still has geometric notions like points or lines or surfaces or ,..... Geometric thinking is surprisingly useful, perhaps because we have a good intuition for it and gives us a parallel understanding to algebra and formalism and a convenient suggestive language.
In functional analysis there are lots of Banach spaces which are not like Euclidean spaces but just have a norm (see also @Andrew Swann's answer).
In the other direction: there are metric spaces which are not like manifolds at all e.g. metrics are used for discrete groups, graphs, or codes (as in coding theory).
Hi Vishesh,
You have a number of interesting answers already. I thought I would add my 2c to see if it also helps you out.
You wrote that:
"The essential feature of a Riemannian Manifold is that it has available a Riemannian Metric Tensor that is defined using an affine connection."
This isn't the case. A Riemannian manifold does indeed always have a Riemannian metric but it is not defined using an affine connection. If the underlying differentiable manifold is nice enough (paracompact and Hausdorff will do the trick) then one can always endow a Riemannian manifold with a canonical compatible and torsion-free connection (the Levi-Civita connection). But the metric doesn't need to be given a-priori: if the underlying manifold is this nice already, then one may always place some Riemannian metric on the manifold.
So, in the end, it is my advice to not couple the metric and connection: view them as distinct objects.
You then wrote that:
"However there are metric spaces that are not inner product spaces.What about them? Do they manifest a manifold like structure or "geometric" structure?"
To this you should definitely look at the answers above. First, not every metric space is a Riemannian manifold. The simplest example (and my favourite) is a cone. It's not smooth enough. Second, take a look at Finsler manifolds. They are the most natural response to your question.
If you have any other questions don't hesitate to ask.
Best,
Glen
@Prof.Wheeler. You are right!! That was a really wrong thing to say. I apologise. In fact the Riemannian Metric is what allows for the Levi-Civita connection to be defined naturally. I put it the other way round. Thanks a lot for pointing out my mistake.
Also, yes I realise that the cone is not a manifold, as the peak or the tip(and consequently a nbd of that) proves to be a hindrance for the smoothness.
Metric geometry is a branch of geometry with metric spaces as the main object of study. It is applied mostly to Riemannian geometry and group theory. Still the theory of Newtonian dynamical systems admitting normal shift of hypersurfaces was first developed for the case of Riemannian manifolds THOUGH it has recently been generalized for manifolds geometric equipment of which is given by some regular Lagrangian or, equivalently, by some regular Hamiltonian dynamical system.
Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set of such points R that the area S(A,B,R) of the triangle ABR vanishes. The triangle area is expressed via metric by means of the Hero's formula, and the straight appears to be defined only via metric, i.e. without a reference to (topological) concept of curve. (Usually the straight is defined as the shortest curve, connecting two points A and B). Such a construction of geometry is free from constraints (continuity, dimensionality of space), generated by a use of topology, but not by geometry in itself. At such a description all information on the geometry properties (such as uniformity, isotropy, continuity and degeneracy) is contained in metric. Modifying the metric, one changes the geometry automatically. The Riemannian geometry is constructed by two different ways: (1) by conventional way on the basis of metric tensor, (2) as a result of modification of the metric in the sigma-immanent description of the proper Euclidean geometry. The nonmetric definition of curve has been shown to be a concept of only proper Euclidean geometry. It is inadequate to any non-Euclidean geometry.
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