The abstract idea of a Riemannian manifold represented as an object with all main functions implemented is one of the core concepts of this toolbox, all further algorithms for example are based on.
To start, you might want to look at the abstract manifold class, for the remaining implementations, this documentation also lists the implemented formulae. For further introductions to Riemannian manifolds or optimization thereon we refer for example to the text books [1,2,3].
Manifolds
 The Hyperbolic Space $\mathbb H^n$

The Riemannian manifold called the hyperbolic space can be isometrically embedded into using the Minkowski inner product.
more  The Euclidean Space $\mathbb R^n$

This class is merely a class for checks on real valued data, which is of course just a special case of manifoldvalued data.
more  The Sphere $\mathbb S^1$

The Sphere can be parametrized by an angle from . While it is also possible to use an embedding into , see the sphere , the amount of data to store is smaller for this special case.
more  The Space $(\mathbb S^1)^m\mathbb R^n$ of combined Cyclic and Vector Space Data

This class is a product manifold of a number of spheres and the Euclidean vector space .
more  The Sphere $\mathbb S^n$

The Sphere can be isometrically embedded into . Geodesics are always great arcs. Take two points , i.e. both are unit vectors . Assume first, they are not antipodal. Then the plane given by the origin, and is unique and we obtain two ways to get from to in this plane; both are great arcs and the shorter one is the shortest path on from to . This situation in the plane is equivalent to looking at the within this plane.
more  The symmetric positive definite $\SPD{n}$ matrices of size $n\times n$

The manifold consists of all matrices such that for all vectors the inequality
holds.
more  The manifold class

The Matlab class
manifold
represents a manifold in general and handles directly also everything as a product manifold using the vector and matrix features of Matlab.All manifolds inherit from manifold, such that both the algorithms and several functions within the manifold are based on the following basic properties a child class should implement.
more
References

Absil, PA, Mahony, R and Sepulchre, R (2008). Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton and Oxford
Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a socalled manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction–illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several wellknown optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The stateoftheart algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra.@book{AMS08, title = {Optimization Algorithms on Matrix Manifolds}, author = {Absil, P.A. and Mahony, R. and Sepulchre, R.}, publisher = {Princeton University Press}, address = {Princeton and Oxford}, year = {2008}, isbn = {9780691132983} }

Carmo, M P do (1992). Riemannian Geometry. Birkhäuser, Basel
Tranlated by F. Flatherty.
Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese) for firstyear graduate students in mathematics and physics. The author’s treatment goes very directly to the basic language of Riemannian geometry and immediately presents some of its most fundamental theorems. It is elementary, assuming only a modest background from readers, making it suitable for a wide variety of students and course structures. Its selection of topics has been deemed "superb" by teachers who have used the text.
A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem. The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student’s understanding and extend knowledge and insight into the subject. Instructors and students alike will find the work to be a significant contribution to this highly applicable and stimulating subject.@book{dCa92, title = {Riemannian Geometry}, author = {do Carmo, Manfredo Perdig{\~a}o}, note = {Tranlated by F.~Flatherty.}, volume = {115}, publisher = {Birkh\"auser}, address = {Basel}, year = {1992} } 
Jost, J (2017). Riemannian Geometry and Geometric Analysis. Springer, Cham
This established reference work continues to provide its readers with a gateway to some of the most interesting developments in contemporary geometry. It offers insight into a wide range of topics, including fundamental concepts of Riemannian geometry, such as geodesics, connections and curvature; the basic models and tools of geometric analysis, such as harmonic functions, forms, mappings, eigenvalues, the Dirac operator and the heat flow method; as well as the most important variational principles of theoretical physics, such as YangMills, GinzburgLandau or the nonlinear sigma model of quantum field theory. The present volume connects all these topics in a systematic geometric framework. At the same time, it equips the reader with the working tools of the field and enables her or him to delve into geometric research.@book{Jos17, author = {Jost, J{\"u}rgen}, title = {Riemannian Geometry and Geometric Analysis}, publisher = {Springer, Cham}, year = {2017}, isbn = {9783319618609}, doi = {10.1007/9783319618609}, series = {Universitext} }