In the belief system of General Relativity imbedding space of course has absolutely no relevance. In TGD framework it is of utmost significance. This is why TGD is revolutionary! Imbedding space defines the "shape" of space-time surface as "seen" by 8-D observer, its symmetries define quantum number spectrum, its spinor connection defines weak gauge fields classically, it brings in many-sheeted space-time, etc. Surface property extends the theory of gravitation to a unified theory of known interactions. The irony is that you get also non-gravitational nteractions by reducing the number of local field degrees of freedom (to four!) rather than introducing a lot of new ones!
Imbeddability to 8-D space-time implies that space-time dynamics is extremely constrained as compared to that in GRT. This has been of course basic worry shadowing the amazing predictions such as understanding of standard model symmetries and quantum numbers and solution of basic problems of cosmology (why mass density is below critical, absence of horizons,…). The ultimate solution of the problem was that the transition to GRT space-time involves in long length scales replacement of many-sheeted - this is the key-word- space-time regions with regions of Minkowski space whose metric is sum of M4 metric and deviations of the metrics of sheets from M4 metric. Similar description applies to gauge potentials and gives standard model gauge fields. In simple situations one can assume that this effective GRT space-time is representable as 4-surface in H and I do this with successful predictions in cosmology.
Imbedding of 2-D string world sheet of course mattered also in the original string model. Unfortunately our space-time is not 2-dimensional but a trick called spontaneous compactification was invented and imbedding space as something given and possibly fixed by the consistency and existence of the theory was replaced with dynamical one. This was a desperate attempt to obtain 4-D space time at least in certain approximation.
The correct manner to proceed would have been as an answer to the question how to generalize string model by replacing string world sheets with 4-D surfaces. Even worse, for "technical reasons" also the idea about string world sheet as sub-manifold with induced metric was given up and dynamical metric at it was induced (there was also the failure to realize that also spinor connection and spinor structure could be induced: this would have immediately led to the geometrization of known interactions and quantum numbers).
As a result, one obtained 2 gravities: stringy 2-D gravity and 10-D gravity to become later 11-D gravity in M-theory context. This from the attempt to quantize 4-D gravity! I must say that I felt deep co-shame and pity since I could only passively inspect as the catastrophe took place without being able to help. This was like seeing helpless victims of a traffic accident to suffer without being able to do anything except to wait for ambulance.
Now a common realization of the horrible mistake has emerged. For reasons easy to understand, this is not a reason for hypeing and the community has wisely chosen to be silent. There are however still some very loudy and very aggressive advocates of the dead science program, who obviously have failed to realize what went wrong (rather amazing but possible if one has a "true fan" attitude).
Perhaps not surprisingly, these individuals are not doing active research on strings themselves (one of the advocates sees everyone disagreeing with him as an evil communist and explains that the reason is that doing research without salary would be communism!). They are trying desperately to gain some scientific respectability using besides ultra-aggressive rhetorics a classical trick: make a lot of noise about pseudoscience since this gives you unearned scientific status. Cold fusion, water memory and anything challenging text book wisdom are excellent targets in this respect. In Finland, people calling for some reason themselves skeptics - most of them actually academic dropouts desperately wanting to appear in the role of wise scientist - are using the same simple trick.
Recipe is simple and conforms with the ultraconservative attitudes of these people explaining also why they are not able to do real research, which always requires challenging of own beliefs. Direct the negative emotions against a researcher, who has found something which does not conform with the text book wisdom. Emotional brain begins to dominate and logical thinking ceases. You indeed get the praise of those who do not have the needed background to decide themselves and are too naive to realize that they are cheated.
Rossi's E-cat used by Lubos for the propagandistic purposes is a perfect target in this respect. I am just wondering how Lubos can see Rossi's E-cat as a failure because it is not commercialized during 4 years and at same time sees no reason to worry about hot fusion program, which has lasted at least 60 years and the outcome is only a bet from Lockheed Martin that fusion plants will be here around 2024. Needless to say, cold fusion plants were here after decade already at seventies and probably already before this!;-)