{'fraction_non_alphanumeric': 0.03148148148148148, 'fraction_numerical': 0.0, 'mean_word_length': 4.75531914893617, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Mean-value theorems for Riemannian manifolds

Synopsis Let Mm (r, f) denote the mean-value of a real-valued integrable function f over a geodesic sphere with centre m and radius r in an n-dimensional Riemannian manifold M. We obtain an expansion of Mm (r, f) in powers of r, thereby generalizing Pizzetti's formula valid in euclidean space. From this expansion we prove that the property for every harmonic function near m, characterizes Einstein spaces. We define super-Einstein spaces and prove that they are characterized by the property