(I) Grössenbegriffe sind nur da möglich, wo sich ein allgemeiner Begriff vorfindet, der verschiedene Bestimmungsweisen zulässt. Je nachdem unter diesen Bestimmungsweisen von einer zu einer andern ein stetiger Uebergang statt- findet oder nicht, bilden sie eine stetige oder discrete Mannigfaltigkeit; die einzelnen Bestimmungsweisen heissen im erstern Falle Punkte, im letztern Elemente dieser Mannigfaltigkeit.
Translation: Concepts of quantity are only possible, where a common concept is found, which allows diffrent ways of determination. Depending on whether under those ways of determination from one to another a continuous transition occurs or not, they form a continous or discrete manifold; the particular ways of determination are called in the first case points, in the last case elements of this manifold.
(II) [...], oder mit andern Worten, man nehme innerhalb der gegebenen Mannigfaltigkeit eine stetige Function des Orts an, un zwar eine solche Function, welche nicht längs eines Theils dieser Mannigfaltigkeit constant ist. Jedes system von Punkten, wo die Function einen constanten Werth hat, bildet dann eine stetige Mannigfaltigkeit von weniger Dimensionen, als die gegebene. Diese Mannigfaltigkeiten gehen bei Aenderung der Function stetig in einander über; man wird daher annehmen können, dass aus einer von ihnen die übrigen hervorgehen, und es wird dies, allgemein zu reden, so geschehen können, dass jeder Punkt in einen bestimmten Punkt der andern übergeht; die Ausnahmsfälle, deren Untersuchung wichtig ist, können hier unberücksichtigt bleiben.
Translation: [...], or in other words, assume a continuous function of location, more precisely a function, which is not constant lengthwise of a part of this manifold. Each system of points, where the function has a constant value, then forms a continuous manifold of fewer dimensions, than the given one. These manifolds continuously transform into each other when the function changes its values; one can therefore assume, that from one of them the others emerge, and this will be, speaking generally, able to happen in such a way, that each point transforms to a certain point in the other; the exceptional cases, which are important to study, can here be left unconsidered.
Above are some excerpts of On the Hypotheses which lie at the Bases of Geometry and my translations, if you think I mistranslated something or you would like to translate some more, feel free to change it wikistyle, but please mention, that it no longer my original version. The English sounds a little strange, but believe me so does the German and I tried to preseve most of its oddities. Some terms are allmost untranslateable, in those cases I will use the German term in the further discussion.
What can be learnt from the above about Riemann's notion of "Mannigfaltigkeit"?
In (I) Riemann defines a "Mannigfaltigkeit" as consisting of the "Bestimmungsweisen" (ways of determination) of a "Größenbegriff" (concept of quantity), with "Bestimmungsweisen" being the points of a "stetige Mannigfaltigkeit" (continuous manifold). The "stetige Mannigfaltigkeit" is not described as being composed from smaller pieces, nor does he mention local flattness. In another, not translated, part he mentions colors and the locations of "Sinngegenstände" (objects of perception) as the only simple concepts giving rise to "stetige Mannigfaltigkeiten". At another point he calls the possible shapes of a spacial figure a "Mannigfaltigkeit".
While the not translated beginning of (II) contains some references to locallity, the translated second part does not. In this part the "Mannigfaltigkeit" is dissected into pieces, also called "Mannigfaltigkeiten". But those pieces do not overlap, but are longitudinal sections. Such a section is a "system of points" with "constant value" for "a function, which is not constant lengthwise" and resembles a level curve. Such level curves are not necessarily manifolds in the modern sense. Riemann seems to be aware of this as he mentions "exceptional cases" which do not transform one-to-one, which "are important to studied", but "can here be left unconsidered".