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The curvature tensor can be expressed in terms of second fundamental form. Therefore bounded curvature is a necessary condition.

Yet injectivity radius has to be bounded below.

These two conditions might be sufficient. If we assume a bit better regularity (say the covariant derivatives of its curvature tensor are in Hoelder class), then this could be proved along the same lines as the Nash embedding theorem.

P.S. In the formulation, you had to say what you mean by "second fundamental form". Most people think it is only defined for hypersurfaces, but you mean a quadratic form on tangent space with values in the normal space.

The curvature tensor can be expressed in terms of second fundamental form. Therefore bounded curvature is a necessary condition.

Yet injectivity radius has to be bounded below.

These two conditions might be sufficient. If we assume a bit better regularity (say a bound on covariant derivatives of the curvature tensor), then this could be proved along the same lines as the Nash embedding theorem.

Postscript. In the formulation, you had to say what you mean by "second fundamental form". Most people think it is only defined for hypersurfaces, but you mean a quadratic form on tangent space with values in the normal space.

The curvature tensor can be expressed in terms of second fundamental form. Therefore bounded curvature is a necessary condition.

Yet injectivity radius has to be bounded below.

These two conditions are sufficient. It seems that this could be proved along the same lines as the Nash embedding theorem.

P.S. In the formulation, you had to say what you mean by "second fundamental form". Most people think it is only defined for hypersurfaces, but you mean a quadratic form on tangent space with values in the normal space.

The curvature tensor can be expressed in terms of second fundamental form. Therefore bounded curvature is a necessary condition.

Yet injectivity radius has to be bounded below.

These two conditions might be sufficient. If we assume a bit better regularity (say the covariant derivatives of its curvature tensor are in Hoelder class), then this could be proved along the same lines as the Nash embedding theorem.

P.S. In the formulation, you had to say what you mean by "second fundamental form". Most people think it is only defined for hypersurfaces, but you mean a quadratic form on tangent space with values in the normal space.

The curvature tensor can be expressed in terms of second fundamental form. Therefore bounded curvature is a necessary condition.

Yet injectivity radius has to be bounded below.

These two conditions are sufficient. It seems that this could be proved along the same lines as the Nash's embedding theorem.

P.S. In the formulation, you had to say what you mean by "second fundamental form". Most people think it is only defined for hypersurfaces, but you mean a quadratic form on tangent space with values in the normal space.

The curvature tensor can be expressed in terms of second fundamental form. Therefore bounded curvature is a necessary condition.

Yet injectivity radius has to be bounded below.

These two conditions are sufficient. It seems that this could be proved along the same lines as the Nash embedding theorem.

P.S. In the formulation, you had to say what you mean by "second fundamental form". Most people think it is only defined for hypersurfaces, but you mean a quadratic form on tangent space with values in the normal space.